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I/Lista 4/Imagens/2021-11-03-15-46-10-image.png create mode 100644 Matemática Discreta I/Lista 4/Imagens/cbdf8dffb6e245ed2f81937b9070c1a9dcc52977.png create mode 100644 Matemática Discreta I/Lista 4/Imagens/e665c17467939fa319f8cd06741fbab4109a1000.png create mode 100644 Matemática Discreta I/Lista 4/Resolução Lista 4.1 - Relações - MD.md create mode 100644 Matemática Discreta I/Lista 4/Resolução Lista 4.2 - Funções - MD.md create mode 100644 Matemática Discreta I/Lista 4/ackermann.c delete mode 100644 Matemática Discreta I/Relações e Funções.md delete mode 100644 Matemática Discreta I/Resolução Lista 2 - MD.md delete mode 100644 Matemática Discreta I/Resolução Lista 3 - MD.md create mode 100644 Matemática Discreta I/Álgebra Booleana.md diff --git a/Algoritmos e Estruturas de Dados I/EP 1/12543033.c b/Algoritmos e Estruturas de Dados I/EP 1/12543033.c deleted file mode 100644 index 837e413..0000000 --- a/Algoritmos e Estruturas de Dados I/EP 1/12543033.c +++ /dev/null @@ -1,192 +0,0 @@ -/*********************************************************************/ -/** ACH2023 - Algoritmos e Estruturas de Dados I **/ -/** EACH-USP - Segundo Semestre de 2021 **/ -/** Turma 04 - Prof. Luciano Antonio Digiampietri **/ -/** **/ -/** EP 1 - Lista Ligada de Produtos **/ -/** **/ -/** Guilherme de Abreu Barreto nUSP: 12543033 **/ -/** **/ -/*********************************************************************/ -#include "listadeprodutos.h" - -PLISTA criarLista(){ - PLISTA res = (PLISTA) malloc(sizeof(LISTADEPRODUTOS)); - int x; - for (x=0;xLISTADELISTAS[x]=(PONT) malloc(sizeof(REGISTRO)); - res->LISTADELISTAS[x]->id=-1; - res->LISTADELISTAS[x]->quantidade=0; - res->LISTADELISTAS[x]->valorUnitario=0; - res->LISTADELISTAS[x]->proxProd=NULL; - } - return res; -} - - -int tamanho(PLISTA l){ - int tam = 0; - int x; - PONT atual; - for (x=0;xLISTADELISTAS[x]->proxProd; - while (atual) { - atual = atual->proxProd; - tam++; - } - } - return tam; -} - -PONT buscarID(PLISTA l, int id){ - int x; - PONT atual; - for (x=0;xLISTADELISTAS[x]->proxProd; - while (atual) { - if (atual->id == id) return atual; - atual = atual->proxProd; - } - } - return NULL; -} - -PONT buscarIDTipo(PLISTA l, int id, int tipo){ - if (tipo<0 || tipo>=NUMTIPOS) return NULL; - PONT atual = l->LISTADELISTAS[tipo]->proxProd; - while (atual) { - if (atual->id == id) return atual; - atual = atual->proxProd; - } - return NULL; -} - -void buscarTudoSobre (PLISTA l, int id, int *tipo, PONT *atual, PONT *ant) { - for (*tipo = 0; *tipo < NUMTIPOS; *tipo = *tipo + 1){ - *ant = l->LISTADELISTAS[*tipo]; - *atual = (*ant)->proxProd; - while (*atual && (*atual)->id != id) { - *ant = *atual; - *atual = (*atual)->proxProd; - } - if (*atual) - return; - } -} - -void exibirLog(PLISTA f){ - int numElementos = tamanho(f); - printf("Log lista [elementos: %i]\n", numElementos); - int x; - PONT atual; - for (x=0;x ", x); - atual = f->LISTADELISTAS[x]->proxProd; - while (atual){ - printf(" [%i;%i;%i;$%i]", atual->id, atual->quantidade, atual->valorUnitario, atual->quantidade*atual->valorUnitario); - atual = atual->proxProd; - } - printf("\n"); - } - printf("\n"); -} - -int consultarValorUnitario(PLISTA l, int id){ - int x; - PONT atual; - for (x=0;xLISTADELISTAS[x]->proxProd; - while (atual) { - if (atual->id == id) return atual->valorUnitario; - atual = atual->proxProd; - } - } - return 0; -} - -/* Anotações quanto a inserção: - -- Uma lista ligada para cada tipo de produto - -- PLISTA: Variável que armazena um ponteiro para a lista [array] de listas [ligadas]. - -- A função necessita proibir a inserção de um novo produto de mesmo identificador que um produto já presente em *qualquer uma* das listas listadas - -- 0 ≤ id ≤ NUMTIPOS -1 - -- quantidade, valor ≥ 0 - -- Se o valor total for igual, inserir antes. -*/ - -bool inserirNovoProduto(PLISTA l, int id, int tipo, int quantidade, int valor){ - PONT p, novo; - - if (id < 0 || tipo < 0 || quantidade < 0 || valor < 0 || buscarID(l, id)) - return false; - - novo = malloc(sizeof(REGISTRO)); - novo->id = id; - novo->quantidade = quantidade; - novo->valorUnitario = valor; - p = l->LISTADELISTAS[tipo]; - - while (p->proxProd - && p->proxProd->valorUnitario * p->proxProd->quantidade < valor * quantidade) - p = p->proxProd; - - novo->proxProd = p->proxProd; - p->proxProd = novo; - return true; -} - -bool removerItensDeUmProduto(PLISTA l, int id, int quantidade){ - int tipo; - PONT atual, ant; - REGISTRO novo; - - if (quantidade <= 0) - return false; - - atual = ant = NULL; - buscarTudoSobre(l, id, &tipo, &atual, &ant); - - if (!atual || quantidade > atual->quantidade) - return false; - - atual->quantidade -= quantidade; - - if (atual->quantidade * atual-> valorUnitario >= ant->quantidade * ant-> valorUnitario) - return true; - - novo = *atual; - ant->proxProd = atual->proxProd; - free(atual); - - if (novo.quantidade == 0) - return true; - return inserirNovoProduto(l, id, tipo, novo.quantidade, novo.valorUnitario); -} - - -bool atualizarValorDoProduto(PLISTA l, int id, int valor){ - int tipo; - PONT atual, ant; - REGISTRO novo; - - if (valor <= 0) - return false; - - atual = ant = NULL; - buscarTudoSobre(l, id, &tipo, &atual, &ant); - - if (!atual) - return false; - - atual->valorUnitario = valor; - novo = *atual; - ant->proxProd = atual->proxProd; - free(atual); - - return inserirNovoProduto(l, id, tipo, novo.quantidade, novo.valorUnitario); -} diff --git a/Algoritmos e Estruturas de Dados I/EP 1/listadeprodutos.c b/Algoritmos e Estruturas de Dados I/EP 1/listadeprodutos.c index 1e96af8..93d1785 100644 --- a/Algoritmos e Estruturas de Dados I/EP 1/listadeprodutos.c +++ b/Algoritmos e Estruturas de Dados I/EP 1/listadeprodutos.c @@ -61,19 +61,6 @@ PONT buscarIDTipo(PLISTA l, int id, int tipo){ return NULL; } -void buscarTudoSobre (PLISTA l, int id, int *tipo, PONT *atual, PONT *ant) { - for (*tipo = 0; *tipo < NUMTIPOS; *tipo = *tipo + 1){ - *ant = l->LISTADELISTAS[*tipo]; - *atual = (*ant)->proxProd; - while (*atual && (*atual)->id != id) { - *ant = *atual; - *atual = (*atual)->proxProd; - } - if (*atual) - return; - } -} - void exibirLog(PLISTA f){ int numElementos = tamanho(f); printf("Log lista [elementos: %i]\n", numElementos); @@ -119,13 +106,41 @@ int consultarValorUnitario(PLISTA l, int id){ - Se o valor total for igual, inserir antes. */ +/* Funções auxiliares */ + +bool buscarTudoSobre (PLISTA l, int id, int *tipo, PONT *atual, PONT *ant) { + for (*tipo = 0; *tipo < NUMTIPOS; *tipo = *tipo + 1){ + *ant = l->LISTADELISTAS[*tipo]; + *atual = (*ant)->proxProd; + while (*atual && (*atual)->id != id) { + *ant = *atual; + *atual = (*atual)->proxProd; + } + if (*atual) + return true; + } + return false; +} + +void apagarRegistro(PONT atual, PONT ant) { + ant->proxProd = atual->proxProd; + free(atual); +} + +/* Funções principais */ + bool inserirNovoProduto(PLISTA l, int id, int tipo, int quantidade, int valor){ PONT p, novo; - if (id < 0 || tipo < 0 || quantidade < 0 || valor < 0 || buscarID(l, id)) + if (id < 1 + || tipo < 0 + || tipo >= NUMTIPOS + || quantidade < 1 + || valor < 1 + || buscarID(l, id)) return false; - novo = malloc(sizeof(*novo)); + novo = malloc(sizeof(REGISTRO)); novo->id = id; novo->quantidade = quantidade; novo->valorUnitario = valor; @@ -141,52 +156,37 @@ bool inserirNovoProduto(PLISTA l, int id, int tipo, int quantidade, int valor){ } bool removerItensDeUmProduto(PLISTA l, int id, int quantidade){ - int tipo; + int tipo, valor; PONT atual, ant; - REGISTRO novo; - if (quantidade <= 0) - return false; - - atual = ant = NULL; - buscarTudoSobre(l, id, &tipo, &atual, &ant); - - if (!atual || quantidade > atual->quantidade) + if (quantidade <= 0 + || !buscarTudoSobre(l, id, &tipo, &atual, &ant) + || quantidade > atual->quantidade) return false; atual->quantidade -= quantidade; + quantidade = atual->quantidade; + valor = atual->valorUnitario; - if (atual->quantidade * atual-> valorUnitario >= ant->quantidade * ant-> valorUnitario) + if (quantidade * valor > ant->quantidade * ant-> valorUnitario) return true; - novo = *atual; - ant->proxProd = atual->proxProd; - free(atual); + apagarRegistro(atual, ant); - if (novo.quantidade == 0) + if (quantidade == 0) return true; - return inserirNovoProduto(l, id, tipo, novo.quantidade, novo.valorUnitario); + return inserirNovoProduto(l, id, tipo, quantidade, valor); } - bool atualizarValorDoProduto(PLISTA l, int id, int valor){ - int tipo; + int tipo, quantidade; PONT atual, ant; - REGISTRO novo; - if (valor <= 0) + if (valor <= 0 || !buscarTudoSobre(l, id, &tipo, &atual, &ant)) return false; - atual = ant = NULL; - buscarTudoSobre(l, id, &tipo, &atual, &ant); + quantidade = atual->quantidade; + apagarRegistro(atual, ant); - if (!atual) - return false; - - atual->valorUnitario = valor; - novo = *atual; - ant->proxProd = atual->proxProd; - free(atual); - - return inserirNovoProduto(l, id, tipo, novo.quantidade, novo.valorUnitario); + return inserirNovoProduto(l, id, tipo, quantidade, valor); } diff --git a/Algoritmos e Estruturas de Dados I/EP 1/output.txt b/Algoritmos e Estruturas de Dados I/EP 1/output.txt new file mode 100644 index 0000000..98bdee2 --- /dev/null +++ b/Algoritmos e Estruturas de Dados I/EP 1/output.txt @@ -0,0 +1,327 @@ +################# INSERINDO ####################### +Log lista [elementos: 0] + #TIPO: 0 -> + #TIPO: 1 -> + #TIPO: 2 -> + #TIPO: 3 -> + #TIPO: 4 -> + #TIPO: 5 -> + #TIPO: 6 -> + #TIPO: 7 -> + #TIPO: 8 -> + #TIPO: 9 -> + +Insercao retornou true (1) +Log lista [elementos: 1] + #TIPO: 0 -> + #TIPO: 1 -> [2;22;23;$506] + #TIPO: 2 -> + #TIPO: 3 -> + #TIPO: 4 -> + #TIPO: 5 -> + #TIPO: 6 -> + #TIPO: 7 -> + #TIPO: 8 -> + #TIPO: 9 -> + +Insercao retornou true (2) +Log lista [elementos: 2] + #TIPO: 0 -> + #TIPO: 1 -> [2;22;23;$506] + #TIPO: 2 -> + #TIPO: 3 -> + #TIPO: 4 -> [4;4;4;$16] + #TIPO: 5 -> + #TIPO: 6 -> + #TIPO: 7 -> + #TIPO: 8 -> + #TIPO: 9 -> + +Insercao retornou true (3) +Log lista [elementos: 3] + #TIPO: 0 -> + #TIPO: 1 -> [6;8;9;$72] [2;22;23;$506] + #TIPO: 2 -> + #TIPO: 3 -> + #TIPO: 4 -> [4;4;4;$16] + #TIPO: 5 -> + #TIPO: 6 -> + #TIPO: 7 -> + #TIPO: 8 -> + #TIPO: 9 -> + +Insercao retornou true (4) +Log lista [elementos: 4] + #TIPO: 0 -> + #TIPO: 1 -> [6;8;9;$72] [3;22;23;$506] [2;22;23;$506] + #TIPO: 2 -> + #TIPO: 3 -> + #TIPO: 4 -> [4;4;4;$16] + #TIPO: 5 -> + #TIPO: 6 -> + #TIPO: 7 -> + #TIPO: 8 -> + #TIPO: 9 -> + +Insercao retornou false (5) +Log lista [elementos: 4] + #TIPO: 0 -> + #TIPO: 1 -> [6;8;9;$72] [3;22;23;$506] [2;22;23;$506] + #TIPO: 2 -> + #TIPO: 3 -> + #TIPO: 4 -> [4;4;4;$16] + #TIPO: 5 -> + #TIPO: 6 -> + #TIPO: 7 -> + #TIPO: 8 -> + #TIPO: 9 -> + +Insercao retornou false (6) +Log lista [elementos: 4] + #TIPO: 0 -> + #TIPO: 1 -> [6;8;9;$72] [3;22;23;$506] [2;22;23;$506] + #TIPO: 2 -> + #TIPO: 3 -> + #TIPO: 4 -> [4;4;4;$16] + #TIPO: 5 -> + #TIPO: 6 -> + #TIPO: 7 -> + #TIPO: 8 -> + #TIPO: 9 -> + +Insercao retornou false (7) +Log lista [elementos: 4] + #TIPO: 0 -> + #TIPO: 1 -> [6;8;9;$72] [3;22;23;$506] [2;22;23;$506] + #TIPO: 2 -> + #TIPO: 3 -> + #TIPO: 4 -> [4;4;4;$16] + #TIPO: 5 -> + #TIPO: 6 -> + #TIPO: 7 -> + #TIPO: 8 -> + #TIPO: 9 -> + +Insercao retornou false (8) +Log lista [elementos: 4] + #TIPO: 0 -> + #TIPO: 1 -> [6;8;9;$72] [3;22;23;$506] [2;22;23;$506] + #TIPO: 2 -> + #TIPO: 3 -> + #TIPO: 4 -> [4;4;4;$16] + #TIPO: 5 -> + #TIPO: 6 -> + #TIPO: 7 -> + #TIPO: 8 -> + #TIPO: 9 -> + +Insercao retornou false (9) +Log lista [elementos: 4] + #TIPO: 0 -> + #TIPO: 1 -> [6;8;9;$72] [3;22;23;$506] [2;22;23;$506] + #TIPO: 2 -> + #TIPO: 3 -> + #TIPO: 4 -> [4;4;4;$16] + #TIPO: 5 -> + #TIPO: 6 -> + #TIPO: 7 -> + #TIPO: 8 -> + #TIPO: 9 -> + +################# REMOVENDO ####################### +Remocao retornou true (1) +Log lista [elementos: 4] + #TIPO: 0 -> + #TIPO: 1 -> [6;8;9;$72] [3;22;23;$506] [2;22;23;$506] + #TIPO: 2 -> + #TIPO: 3 -> + #TIPO: 4 -> [4;3;4;$12] + #TIPO: 5 -> + #TIPO: 6 -> + #TIPO: 7 -> + #TIPO: 8 -> + #TIPO: 9 -> + +Remocao retornou true (2) +Log lista [elementos: 4] + #TIPO: 0 -> + #TIPO: 1 -> [6;8;9;$72] [2;21;23;$483] [3;22;23;$506] + #TIPO: 2 -> + #TIPO: 3 -> + #TIPO: 4 -> [4;3;4;$12] + #TIPO: 5 -> + #TIPO: 6 -> + #TIPO: 7 -> + #TIPO: 8 -> + #TIPO: 9 -> + +Remocao retornou true (3) +Log lista [elementos: 3] + #TIPO: 0 -> + #TIPO: 1 -> [6;8;9;$72] [2;21;23;$483] + #TIPO: 2 -> + #TIPO: 3 -> + #TIPO: 4 -> [4;3;4;$12] + #TIPO: 5 -> + #TIPO: 6 -> + #TIPO: 7 -> + #TIPO: 8 -> + #TIPO: 9 -> + +Remocao retornou false (4) +Log lista [elementos: 3] + #TIPO: 0 -> + #TIPO: 1 -> [6;8;9;$72] [2;21;23;$483] + #TIPO: 2 -> + #TIPO: 3 -> + #TIPO: 4 -> [4;3;4;$12] + #TIPO: 5 -> + #TIPO: 6 -> + #TIPO: 7 -> + #TIPO: 8 -> + #TIPO: 9 -> + +Remocao retornou false (5) +Log lista [elementos: 3] + #TIPO: 0 -> + #TIPO: 1 -> [6;8;9;$72] [2;21;23;$483] + #TIPO: 2 -> + #TIPO: 3 -> + #TIPO: 4 -> [4;3;4;$12] + #TIPO: 5 -> + #TIPO: 6 -> + #TIPO: 7 -> + #TIPO: 8 -> + #TIPO: 9 -> + +Remocao retornou false (6) +Log lista [elementos: 3] + #TIPO: 0 -> + #TIPO: 1 -> [6;8;9;$72] [2;21;23;$483] + #TIPO: 2 -> + #TIPO: 3 -> + #TIPO: 4 -> [4;3;4;$12] + #TIPO: 5 -> + #TIPO: 6 -> + #TIPO: 7 -> + #TIPO: 8 -> + #TIPO: 9 -> + +################# ATUALIZANDO VALOR ####### +Atualizacao retornou true (1) +Log lista [elementos: 3] + #TIPO: 0 -> + #TIPO: 1 -> [6;8;1;$8] [2;21;23;$483] + #TIPO: 2 -> + #TIPO: 3 -> + #TIPO: 4 -> [4;3;4;$12] + #TIPO: 5 -> + #TIPO: 6 -> + #TIPO: 7 -> + #TIPO: 8 -> + #TIPO: 9 -> + +Atualizacao retornou true (2) +Log lista [elementos: 3] + #TIPO: 0 -> + #TIPO: 1 -> [2;21;23;$483] [6;8;600;$4800] + #TIPO: 2 -> + #TIPO: 3 -> + #TIPO: 4 -> [4;3;4;$12] + #TIPO: 5 -> + #TIPO: 6 -> + #TIPO: 7 -> + #TIPO: 8 -> + #TIPO: 9 -> + +Atualizacao retornou true (3) +Log lista [elementos: 3] + #TIPO: 0 -> + #TIPO: 1 -> [6;8;20;$160] [2;21;23;$483] + #TIPO: 2 -> + #TIPO: 3 -> + #TIPO: 4 -> [4;3;4;$12] + #TIPO: 5 -> + #TIPO: 6 -> + #TIPO: 7 -> + #TIPO: 8 -> + #TIPO: 9 -> + +Atualizacao retornou false (4) +Log lista [elementos: 3] + #TIPO: 0 -> + #TIPO: 1 -> [6;8;20;$160] [2;21;23;$483] + #TIPO: 2 -> + #TIPO: 3 -> + #TIPO: 4 -> [4;3;4;$12] + #TIPO: 5 -> + #TIPO: 6 -> + #TIPO: 7 -> + #TIPO: 8 -> + #TIPO: 9 -> + +Atualizacao retornou false (5) +Log lista [elementos: 3] + #TIPO: 0 -> + #TIPO: 1 -> [6;8;20;$160] [2;21;23;$483] + #TIPO: 2 -> + #TIPO: 3 -> + #TIPO: 4 -> [4;3;4;$12] + #TIPO: 5 -> + #TIPO: 6 -> + #TIPO: 7 -> + #TIPO: 8 -> + #TIPO: 9 -> + +Atualizacao retornou false (6) +Log lista [elementos: 3] + #TIPO: 0 -> + #TIPO: 1 -> [6;8;20;$160] [2;21;23;$483] + #TIPO: 2 -> + #TIPO: 3 -> + #TIPO: 4 -> [4;3;4;$12] + #TIPO: 5 -> + #TIPO: 6 -> + #TIPO: 7 -> + #TIPO: 8 -> + #TIPO: 9 -> + +Atualizacao retornou true (7) +Log lista [elementos: 3] + #TIPO: 0 -> + #TIPO: 1 -> [6;8;3;$24] [2;21;23;$483] + #TIPO: 2 -> + #TIPO: 3 -> + #TIPO: 4 -> [4;3;4;$12] + #TIPO: 5 -> + #TIPO: 6 -> + #TIPO: 7 -> + #TIPO: 8 -> + #TIPO: 9 -> + +Atualizacao retornou true (8) +Log lista [elementos: 3] + #TIPO: 0 -> + #TIPO: 1 -> [6;8;3;$24] [2;21;23;$483] + #TIPO: 2 -> + #TIPO: 3 -> + #TIPO: 4 -> [4;3;200;$600] + #TIPO: 5 -> + #TIPO: 6 -> + #TIPO: 7 -> + #TIPO: 8 -> + #TIPO: 9 -> + +Atualizacao retornou true (9) +Log lista [elementos: 3] + #TIPO: 0 -> + #TIPO: 1 -> [6;8;3;$24] [2;21;20;$420] + #TIPO: 2 -> + #TIPO: 3 -> + #TIPO: 4 -> [4;3;200;$600] + #TIPO: 5 -> + #TIPO: 6 -> + #TIPO: 7 -> + #TIPO: 8 -> + #TIPO: 9 -> + diff --git a/Algoritmos e Estruturas de Dados I/EP 1/usaListaDeProdutos_2.c b/Algoritmos e Estruturas de Dados I/EP 1/usaListaDeProdutos_2.c new file mode 100644 index 0000000..2a0ffa5 --- /dev/null +++ b/Algoritmos e Estruturas de Dados I/EP 1/usaListaDeProdutos_2.c @@ -0,0 +1,39 @@ +#include "listadeprodutos.c" + +int main() { + PLISTA f = criarLista(); + bool res; + + exibirLog(f); + res = inserirNovoProduto(f, 3, 3, 3, 3); + if(res) printf("Insercao retornou true (1)\n"); + else printf("Insercao retornou false (1)\n"); + exibirLog(f); + + res = inserirNovoProduto(f, 1, 2, 3, 5); + if(res) printf("Insercao retornou true (1)\n"); + else printf("Insercao retornou false (1)\n"); + exibirLog(f); + + res = inserirNovoProduto(f, 5, 2, 7, 8); + if(res) printf("Insercao retornou true (1)\n"); + else printf("Insercao retornou false (1)\n"); + exibirLog(f); + + res = inserirNovoProduto(f, 9, 2, 11, 12); + if(res) printf("Insercao retornou true (1)\n"); + else printf("Insercao retornou false (1)\n"); + exibirLog(f); + + res = removerItensDeUmProduto(f, 1, 3); + if(res) printf("Remoção retornou true (1)\n"); + else printf("Remoção retornou false (1)\n"); + exibirLog(f); + + res = removerItensDeUmProduto(f, 9, 11); + if(res) printf("Remoção retornou true (1)\n"); + else printf("Remoção retornou false (1)\n"); + exibirLog(f); + + return 0; +} diff --git a/Algoritmos e Estruturas de Dados I/Semana 03/sorts.c b/Algoritmos e Estruturas de Dados I/Semana 03/sorts.c index cc7d748..3cbc11c 100644 --- a/Algoritmos e Estruturas de Dados I/Semana 03/sorts.c +++ b/Algoritmos e Estruturas de Dados I/Semana 03/sorts.c @@ -1,4 +1,7 @@ #include +#define array int* + +/* Bubble Sort */ void swap (int *a, int *b) { int tmp = *a; @@ -6,78 +9,124 @@ void swap (int *a, int *b) { *b = tmp; } -int *merge (int *array1, int size1, int *array2, int size2) { - int k, i, j, *array3 = malloc((size1 + size2) * sizeof(*array3)); - - for (k = i = j = 0; k < size1 + size2; k++) - array3[k] = ((i < size1 && array1[i] <= array2[j]) || j >= size2) ? array1[i++] : array2[j++]; - return array3; -} - -int partition (int *array, int size) { - int i = -1, j; - - for (j = 0; j < size - 1; j++) - if (array[j] <= array[size - 1]) - swap(&array[++i], &array[j]); - swap(&array[++i], &array[j]); - - return i; -} - -void bubbleSort (int *array, int size) { +void bubbleSort (array A, int size) { int i; if (size <= 1) return; for (i = 0; i < size; i++) - if (array[i] > array[i + 1]) - swap(&array[i], &array[i + 1]); - bubbleSort(array, size - 1); + if (A[i] > A[i + 1]) + swap(&A[i], &A[i + 1]); + bubbleSort(A, size - 1); } -void selectionSort (int *array, int size) { - int i, pos = 0; +/* Selection Sort */ - if (size <= 1) - return; - for (i = 1; i < size; i++) - if (array[i] < array[pos]) +int * minimalValue (array A, int size) { + int i, pos = 0; + + for (i = 1; i < size; i++) + if (A[i] < A[pos]) pos = i; - swap(array, &array[pos]); - selectionSort(array + 1, size - 1); + return A + pos; } -void insertionSort (int *array, int size) { +void selectionSort (array A, int size) { + swap(A, minimalValue(A, size)); + selectionSort(A + 1, size - 1); +} + +/* Insertion Sort */ + +void insertionSort (array A, int size) { int i, tmp; if (size <= 1) return; - insertionSort(array, size - 1); - tmp = array[size - 1]; - for (i = size - 2; i >= 0 && array[i] > tmp; i--) - array[i + 1] = array[i]; - array[i + 1] = tmp; + insertionSort(A, --size); + tmp = A[size--]; + for (i = size; i >= 0 && A[i] > tmp; i--) + A[i + 1] = A[i]; + A[i + 1] = tmp; } -void mergeSort(int *array, int size) { - int *tmp, i, m = size / 2; +/* Merge Sort */ - if (size > 1) { - mergeSort(array, m); - mergeSort(array + m, size - m); - } - tmp = merge(array, m, array + m, size - m); - for (i = 0; i < size; i++) - array[i] = tmp[i]; +void merge (array A, int pivot, int size) { + int i, k, j = pivot; + array tmp = malloc(size * sizeof(int)); + + for (i = k = 0; k < size; k++) + tmp[k] = ((A[i] <= A[j] && i < pivot) || j == size) ? + A[i++] : A[j++]; + for (k = 0; k < size; k++) + A[k] = tmp[k]; + free(tmp); } -void quickSort (int *array, int size) { +void mergeSort (array A, int size) { int pivot; if (size <= 1) return; - pivot = partition(array, size); - quickSort(array, pivot); - quickSort(array + (pivot + 1), size - (pivot + 1)); + pivot = size / 2; + mergeSort(A, pivot); + mergeSort(A + pivot, size - pivot); + merge(A, pivot, size); +} + + +/* Quick Sort */ + +int partition (array A, int size) { + int i, j, lastIndex = size - 1; + + for (i = j = 0; i < lastIndex; i++) + if (A[i] <= A[lastIndex]) + swap(&A[i], &A[j++]); + swap(&A[i], &A[j]); + return j; +} + +void quickSort (array A, int size) { + int pivot; + + if (size <= 1) + return; + pivot = partition(A, size); + quickSort(A, pivot++); + quickSort(A + pivot, size - pivot); +} + +/* Heap Sort */ + +void heapify (array A, int size, int i) { + int max = i, left = 2 * i + 1, right = left + 1; + + if (left < size && A[left] > A[max]) + max = left; + + if (right < size && A[right] > A[max]) + max = right; + + if (max == i) + return; + + swap(A + i, A + max); + heapify(A, size, max); +} + +void heapSort (array A, int size) { + int i; + + if (size <= 1) + return; + + for (i = (size - 1) / 2; i <= 0; i--) + heapify(A, size, i); + + for (i = size - 1; i >= 0; i--) { + swap(A, A + i); + heapify(A, i, 0); + } } diff --git a/Algoritmos e Estruturas de Dados I/Semana 04/Semana 04.md b/Algoritmos e Estruturas de Dados I/Semana 04/Semana 04.md index cd36748..8ad8513 100644 --- a/Algoritmos e Estruturas de Dados I/Semana 04/Semana 04.md +++ b/Algoritmos e Estruturas de Dados I/Semana 04/Semana 04.md @@ -7,134 +7,133 @@ #include typedef struct node { - int value; - struct node *prev; - struct node *next; + int value; + struct node *prev; + struct node *next; } Node; typedef struct { - Node *start; - Node *end; + Node *start; + Node *end; } List; void insertIn (List *l, int value) { - Node *prev, *current, *next; - prev = next = NULL; + Node *prev, *current, *next; + prev = next = NULL; - if (l->start != NULL) { - if (abs(value - l->start->value) < abs(value - l->end->value)) { - for (next = l->start; next != NULL && next->value < value; next = next->next) - prev = next; - if (next != NULL && next->value == value) - return; - } - else { - for (prev = l->end; prev != NULL && prev->value > value; prev = prev->prev) - next = prev; - if (prev != NULL && prev->value == value) - return; - } - } + if (l->start != NULL) { + if (abs(value - l->start->value) < abs(value - l->end->value)) { + for (next = l->start; next != NULL && next->value < value; next = next->next) + prev = next; + if (next != NULL && next->value == value) + return; + } + else { + for (prev = l->end; prev != NULL && prev->value > value; prev = prev->prev) + next = prev; + if (prev != NULL && prev->value == value) + return; + } + } - current = malloc(sizeof(*current)); - current->value = value; + current = malloc(sizeof(*current)); + current->value = value; - if (prev == NULL) { - current->next = l->start; - l->start = current; - current->prev = NULL; - } - else { - current->next = prev->next; - prev->next = current; - } + if (prev == NULL) { + current->next = l->start; + l->start = current; + current->prev = NULL; + } + else { + current->next = prev->next; + prev->next = current; + } - if (next == NULL) { - current->prev = l->end; - l->end = current; - current->next = NULL; - } - else { - current->prev = next->prev; - next->prev = current; - } + if (next == NULL) { + current->prev = l->end; + l->end = current; + current->next = NULL; + } + else { + current->prev = next->prev; + next->prev = current; + } } void removeFrom (List *l, int value) { - Node *current; + Node *current; - if (l->start == NULL) - return; - if (abs(value - l->start->value) < abs(value - l->end->value)) { - current = l->start; - while (current != NULL && current->value < value) - current = current->next; - } - else { - current = l->end; - while (current != NULL && current->value > value) - current = current->prev; - } + if (l->start == NULL) + return; + if (abs(value - l->start->value) < abs(value - l->end->value)) { + current = l->start; + while (current != NULL && current->value < value) + current = current->next; + } + else { + current = l->end; + while (current != NULL && current->value > value) + current = current->prev; + } - if (current == NULL) - return; - if (current->prev != NULL && current->next != NULL) { - current->prev->next = current->next; - current->next->prev = current->prev; - } - else { - if (current->prev == NULL) { - l->start = current->next; - if (current->next != NULL) - current->next->prev = NULL; - } - if (current->next == NULL) { - l->end = current->prev; - if (current->prev != NULL) - current->prev->next = NULL; - } - } - free(current); + if (current == NULL) + return; + if (current->prev != NULL && current->next != NULL) { + current->prev->next = current->next; + current->next->prev = current->prev; + } + else { + if (current->prev == NULL) { + l->start = current->next; + if (current->next != NULL) + current->next->prev = NULL; + } + if (current->next == NULL) { + l->end = current->prev; + if (current->prev != NULL) + current->prev->next = NULL; + } + } + free(current); } void printList (List l) { - int i; - Node *current = l.start; + int i; + Node *current = l.start; - for (i = 0; current != NULL; i++) { - printf("\nÍndice: %-6d Valor: %-6d (Endereços) Anterior: %p Atual: %p Próximo: %p\n", i, current->value, current->prev, current, current->next); - current = current->next; - } + for (i = 0; current != NULL; i++) { + printf("\nÍndice: %-6d Valor: %-6d (Endereços) Anterior: %p Atual: %p Próximo: %p\n", i, current->value, current->prev, current, current->next); + current = current->next; + } } int main () { - int i; - char c; + int i; + char c; List l = {0}; - printf("Este programa adiciona valores inteiros à uma lista duplamente ligada, e depois permite excluí-los da mesma.\nDigite uma sequência de valores e pressione ENTER: "); - do { - if(!scanf("%d%c", &i, &c)) { - printf("Valor inválido detectado."); - return 1; - } - insertIn(&l, i); - } while (c != '\n'); - printList(l); + printf("Este programa adiciona valores inteiros à uma lista duplamente ligada, e depois permite excluí-los da mesma.\nDigite uma sequência de valores e pressione ENTER: "); + do { + if(!scanf("%d%c", &i, &c)) { + printf("Valor inválido detectado."); + return 1; + } + insertIn(&l, i); + } while (c != '\n'); + printList(l); - printf("\nDigite uma sequência de valores a serem apagados e pressione ENTER: "); - do { - if(!scanf("%d%c", &i, &c)) { - printf("Valor inválido detectado."); - return 1; - } - removeFrom(&l, i); - } while (c != '\n'); - printList(l); + printf("\nDigite uma sequência de valores a serem apagados e pressione ENTER: "); + do { + if(!scanf("%d%c", &i, &c)) { + printf("Valor inválido detectado."); + return 1; + } + removeFrom(&l, i); + } while (c != '\n'); + printList(l); return 0; } - ``` [^1]: nUSP 12543033 \ No newline at end of file diff --git a/Algoritmos e Estruturas de Dados I/Semana 09/MatrizDinamica2.c b/Algoritmos e Estruturas de Dados I/Semana 09/MatrizDinamica2.c new file mode 100644 index 0000000..85fa5c6 --- /dev/null +++ b/Algoritmos e Estruturas de Dados I/Semana 09/MatrizDinamica2.c @@ -0,0 +1,83 @@ +#include +#include + +typedef struct { + int linhas; + int colunas; + int** M; +}MATRIZ; + +MATRIZ* inicializarMatriz(int linhas, int colunas){ + if (linhas<=0 || colunas<=0) return NULL; + int i; + MATRIZ* mat = (MATRIZ*) malloc(sizeof(MATRIZ)); + mat->M = (int**)malloc(sizeof(int*)*linhas); + for (i=0; iM[i] = (int*)malloc(sizeof(int)*colunas); + } + mat->linhas = linhas; + mat->colunas = colunas; + return mat; +} + +void imprimirMatriz(MATRIZ* mat){ + int i,j; + printf("Matriz %i x %i:\n", mat->linhas, mat->colunas); + for (i=0; ilinhas; i++) { + for (j=0; jcolunas; j++) printf("%i ", mat->M[i][j]); + printf("\n"); + } + printf("\n"); +} + +MATRIZ *multiplicarMatrizes(MATRIZ *m1, MATRIZ *m2){ + int i,j,k; + MATRIZ *res; + + if (m1->colunas != m2->linhas) + return NULL; + res = inicializarMatriz(m1->linhas, m2->colunas); + for (i = 0; i < res->linhas; i++) { + for (j = 0; j < res->colunas; j++) { + res->M[i][j] = 0; + for (k = 0; k < m1->colunas; k++) + res->M[i][j] += m1->M[i][k] * m2->M[k][j]; + } + } + return res; +} + + +int main(){ + + MATRIZ* m1 = inicializarMatriz(3, 2); + int i,j; + int cont = 1; + for (i=0; ilinhas; i++) + for (j=0; jcolunas; j++) { + m1->M[i][j] = cont; + cont++; + } + + imprimirMatriz(m1); + + MATRIZ* m2 = inicializarMatriz(2, 3); + for (i=0; ilinhas; i++) + for (j=0; jcolunas; j++) { + m2->M[i][j] = cont; + cont++; + } + + imprimirMatriz(m2); + + MATRIZ* m3 = multiplicarMatrizes(m1, m2); + + imprimirMatriz(m3); + + + char c1; + printf("Pressione ENTER para terminar.\n"); + scanf("%c", &c1); + + return 0; +} diff --git a/Algoritmos e Estruturas de Dados I/Semana 09/Semana 09.md b/Algoritmos e Estruturas de Dados I/Semana 09/Semana 09.md new file mode 100644 index 0000000..6f3e2a1 --- /dev/null +++ b/Algoritmos e Estruturas de Dados I/Semana 09/Semana 09.md @@ -0,0 +1,24 @@ +# Semana 09: resposta ao [exercício proposto](http://www.each.usp.br/digiampietri/ACH2023/ACH2023_AtividadeSemanal09.pdf) + +> Por Guilherme de Abreu Barreto[^1] + +```c +MATRIZ *multiplicarMatrizes(MATRIZ *m1, MATRIZ *m2){ + int i,j,k; + MATRIZ *res; + + if (m1->colunas != m2->linhas) + return NULL; + res = inicializarMatriz(m1->linhas, m2->colunas); + for (i = 0; i < res->linhas; i++) { + for (j = 0; j < res->colunas; j++) { + res->M[i][j] = 0; + for (k = 0; k < m1->colunas; k++) + res->M[i][j] += m1->M[i][k] * m2->M[k][j]; + } + } + return res; +} +``` + +[^1]: nUSP: 12543033 \ No newline at end of file diff --git a/Algoritmos e Estruturas de Dados I/Semana 09/saidaMatriz.txt b/Algoritmos e Estruturas de Dados I/Semana 09/saidaMatriz.txt new file mode 100644 index 0000000..92e5dc6 --- /dev/null +++ b/Algoritmos e Estruturas de Dados I/Semana 09/saidaMatriz.txt @@ -0,0 +1,15 @@ +Matriz 3 x 2: +1 2 +3 4 +5 6 + +Matriz 2 x 3: +7 8 9 +10 11 12 + +Matriz 3 x 3: +27 30 33 +61 68 75 +95 106 117 + +Pressione ENTER para terminar. diff --git a/Algoritmos e Estruturas de Dados I/Semana 10/Semana 10.md b/Algoritmos e Estruturas de Dados I/Semana 10/Semana 10.md new file mode 100644 index 0000000..e53ff93 --- /dev/null +++ b/Algoritmos e Estruturas de Dados I/Semana 10/Semana 10.md @@ -0,0 +1,55 @@ +# Semana 10: resposta ao [exercício proposto](http://www.each.usp.br/digiampietri/ACH2023/ACH2023_AtividadeSemanal10.pdf) + +> Por Guilherme de Abreu Barreto[^1] + +**Parte 1:** Escreva o que será impresso pela execução do código desta atividade. + +**Resolução:** + +```c +FuncaoZZZ (1a execucao): 1 +FuncaoX (1a execucao): 3 +Imprimindo (1a execucao): 3 2 1 +FuncaoZZZ (2a execucao): 3 +FuncaoX (2a execucao): 6 +Imprimindo (2a execucao): 3 7 6 5 2 1 +``` + +**Parte 2:** Desenhe a arvore binária resultante da execução deste código. + +**Resolução:** + +```mermaid +classDiagram +Raiz --> esq1 +Raiz --> dir1 +dir1 --> esq2 +dir1 --> dir2 +esq2 --> dir3 +class Raiz { + (PONT) 0x5555555592a0 + chave = 1 +} +class esq1 { + chave = 3 + (PONT) 0x5555555592e0 +} +class dir1 { + chave = 2 + (PONT) 0x5555555592c0 +} +class esq2 { + chave = 6 + (PONT) 0x555555559750 +} +class dir2 { + chave = 5 + (PONT) 0x555555559730 +} +class dir3 { + chave = 7 + (PONT) 0x555555559710 +} +``` + +[^1]: nUSP: 12543033 \ No newline at end of file diff --git a/Algoritmos e Estruturas de Dados I/Semana 10/arvoresBinaria2.c b/Algoritmos e Estruturas de Dados I/Semana 10/arvoresBinaria2.c new file mode 100644 index 0000000..ef052d2 --- /dev/null +++ b/Algoritmos e Estruturas de Dados I/Semana 10/arvoresBinaria2.c @@ -0,0 +1,103 @@ +#include +#include +#define true 1 +#define false 0 +typedef enum{esquerdo,direito} LADO; +typedef int bool; +typedef int TIPOCHAVE; + +typedef struct aux{ + TIPOCHAVE chave; + struct aux *esq, *dir; +} NO, *PONT; + +PONT buscarChave(TIPOCHAVE ch, PONT raiz){ + if (raiz == NULL) return NULL; + if (raiz->chave == ch) return raiz; + PONT aux = buscarChave(ch,raiz->esq); + if (aux) return aux; + return buscarChave(ch,raiz->dir); +} + +void apagar(PONT raiz){ + if (!raiz) return; + apagar(raiz->esq); + apagar(raiz->dir); + free(raiz); +} + +PONT criarNovoNo(TIPOCHAVE ch){ + PONT novoNo = (PONT)malloc(sizeof(NO)); + novoNo->esq = novoNo->dir = NULL; + novoNo->chave = ch; + return novoNo; +} + +bool inserirFilho(PONT raiz, TIPOCHAVE novaChave, TIPOCHAVE chavePai, LADO lado){ + PONT pai = buscarChave(chavePai,raiz); + if (!pai) return false; + PONT novo = criarNovoNo(novaChave); + if (lado == esquerdo){ + apagar(pai->esq); + pai->esq = novo; + }else{ + apagar(pai->dir); + pai->dir = novo; + } + return true; +} + +void exibirArvoreOrdemW(PONT raiz){ + if (raiz == NULL) return; + exibirArvoreOrdemW(raiz->esq); + exibirArvoreOrdemW(raiz->dir); + printf("%i ",raiz->chave); +} + +int max(int a, int b){ + if (a>b) return a; + return b; +} + +int funcaoZZZ(PONT raiz){ + if (!raiz ) return -1; + return 1 + max(funcaoZZZ(raiz->esq), funcaoZZZ(raiz->dir)); +} + +int funcaoX(PONT raiz){ + if (!raiz ) return 0; + return 1 + funcaoX(raiz->esq) + funcaoX(raiz->dir); +} + +void inicializar(PONT* raiz){ + *raiz = NULL; +} + +void criarRaiz(PONT* raiz, TIPOCHAVE novaChave){ + *raiz = criarNovoNo(novaChave); +} + + +int main(){ + PONT raiz; + inicializar(&raiz); + criarRaiz(&raiz,1); + inserirFilho(raiz,2,1,direito); + inserirFilho(raiz,3,1,esquerdo); + printf("FuncaoZZZ (1a execucao): %i\n",funcaoZZZ(raiz)); + printf("FuncaoX (1a execucao): %i\n",funcaoX(raiz)); + printf("Imprimindo (1a execucao): "); + exibirArvoreOrdemW(raiz); + printf("\n"); + + inserirFilho(raiz,4,2,esquerdo); + inserirFilho(raiz,5,2,direito); + inserirFilho(raiz,6,2,esquerdo); + inserirFilho(raiz,7,6,direito); + printf("FuncaoZZZ (2a execucao): %i\n",funcaoZZZ(raiz)); + printf("FuncaoX (2a execucao): %i\n",funcaoX(raiz)); + printf("Imprimindo (2a execucao): "); + exibirArvoreOrdemW(raiz); + printf("\n"); + return 0; +} diff --git a/Algoritmos e Estruturas de Dados I/Semana 10/binary_tree.c b/Algoritmos e Estruturas de Dados I/Semana 10/binary_tree.c new file mode 100644 index 0000000..9511033 --- /dev/null +++ b/Algoritmos e Estruturas de Dados I/Semana 10/binary_tree.c @@ -0,0 +1,45 @@ +#include +#include + +typedef enum {left, right} side; + +typedef struct node{ + int value; + struct node *left, *right; +} Node; + +Node * findValue (Node *parent, int value) { + Node * tmp; + + if (!parent || parent->value == value) + return parent; + tmp = findValue(parent->left, value); + return (tmp) ? tmp : findValue(parent->right, value); +} + +Node * newNode (int value) { + Node *new = malloc(sizeof(Node)); + new->value = value; + new->left = new->right = NULL; + return new; +} + +void createRoot (Node **root, int value) { + *root = newNode(value); +} + +bool setChild (Node * root, int childValue, int parentValue, side s) { + Node *child, *parent = findValue(root, parentValue); + if (!parent) + return false; + child = newNode(childValue); + if (s == left) { + child->left = parent->left; + parent->left = child; + } + else { + child->right = parent->right; + parent->right = child; + } + return true; +} diff --git a/Algoritmos e Estruturas de Dados I/Semana 11/binary_tree.c b/Algoritmos e Estruturas de Dados I/Semana 11/binary_tree.c new file mode 100644 index 0000000..89d249b --- /dev/null +++ b/Algoritmos e Estruturas de Dados I/Semana 11/binary_tree.c @@ -0,0 +1,108 @@ +#include +#include + +typedef struct node{ + int value; + + /* Pointers to nodes which store values that are less or equal (le) and greater than (gt) the current node's. */ + struct node *parent, *le, *gt; +} Node; + +Node * newNode(int value){ + Node *new = malloc(sizeof(Node)); + new->parent = new->le = new->gt = NULL; + new->value = value; + return new; +} + +bool insertNode(Node *root, Node *new) { + Node *branch; + + if (!new) + return false; + if (!root) { + root = new; + return true; + } + do { + branch = (new->value > root->value) ? root->gt : root->le; + if (branch) + root = branch; + } while (branch); + + branch = new; + new->parent = root; + return true; +} + +Node * findNode(Node *root, int value) { + if (!root) + return NULL; + if (root->value == value) + return root; + if (root->value > value) + return findNode(root->le, value); + return findNode(root->gt, value); +} + +Node * findMaxValue (Node *root) { + while (root->gt) + root = root->gt; + return root; +} + +void rearrangeTree (Node *root, Node *branch) { + if (branch->parent != root) { + branch->parent->gt = branch->le; + branch->le = root->le; + } + branch->gt = root->gt; +} + +Node * selectBranch (Node *root) { + Node *branch; + + if (root->le) { + if (!root->gt) + return root->le; + branch = findMaxValue(root->le); + rearrangeTree(root, branch); + return branch; + } + return root->gt; +} + +void substituteNode (Node *n, Node *branch) { + if (n->parent) { + if (n->value < n->parent->value) + n->parent->le = branch; + else + n->parent->gt = branch; + branch->parent = n->parent; + } + else + branch->parent = NULL; + free(n); + n = branch; +} + +Node * excludeNode(Node *root, int value) { + Node *n = findNode(root, value); + + if (!n) + return n; + substituteNode(n, selectBranch(n)); + return (n->parent) ? root : n; +} + +int treeSize (Node *root) { + if (!root) + return 0; + return treeSize(root->le) + treeSize(root->gt) + 1; +} + +int main () { + Node *new = newNode(23), *r = NULL; + insertNode(r,new); + return 0; +} diff --git a/BxComp b/BxComp index 6a04e7a..d4a6592 160000 --- a/BxComp +++ b/BxComp @@ -1 +1 @@ -Subproject commit 6a04e7a8fac2a8633a57908becda0e30f216dcc7 +Subproject commit d4a659215ec8a9097e0bf8d670200112afd92dc3 diff --git a/Cálculo II/Atividade 5/Atividade 5.md b/Cálculo II/Atividade 5/Atividade 5.md index d878918..c49c805 100644 --- a/Cálculo II/Atividade 5/Atividade 5.md +++ b/Cálculo II/Atividade 5/Atividade 5.md @@ -1,6 +1,6 @@ # Atividade 5 -Resolução dos exercícios obrigatórios, feita por Guilherme de Abreu Barreto[^1]. +> Resolução dos exercícios obrigatórios, feita por Guilherme de Abreu Barreto[^1]. ## Capítulo 12.1 diff --git a/Cálculo II/Atividade 5/Vetores.md b/Cálculo II/Atividade 5/Vetores.md index e22029d..8b8d721 100644 --- a/Cálculo II/Atividade 5/Vetores.md +++ b/Cálculo II/Atividade 5/Vetores.md @@ -54,6 +54,4 @@ O comprimento de um vetor tridimensional $\textbf a = \langle a_1,a_2, a_3 \rang $$ |\textbf a | = \sqrt{a_1^2 + a_2^2 + a_3^2} -$$ - - +$$ \ No newline at end of file diff --git a/Cálculo II/Atividade 6/Atividade 6.md b/Cálculo II/Atividade 6/Atividade 6.md new file mode 100644 index 0000000..54b3f78 --- /dev/null +++ b/Cálculo II/Atividade 6/Atividade 6.md @@ -0,0 +1,105 @@ +# Atividade 6 + +> Resolução dos exercícios obrigatórios, feita por Guilherme de Abreu Barreto[^1]. + +## Capítulo 12.3 + +### Exercício 39 + +Determine o vetor projeção e a projeção escalar de **b** sobre **a** onde + +$$ +\textbf a = \lang -5,12 \rang;\, \textbf b = \lang 4,6\rang +$$ + +### Resolução + +$\text{comp}_\textbf a \textbf b = \dfrac{\textbf a \cdot \textbf b}{|\textbf a|} = \dfrac{-5 \cdot 4 + 12 \cdot 6}{\sqrt{(-5)^2 + 12^2}} = 4$ + +$\text{proj}_\textbf a \textbf b = \dfrac{\textbf a}{|\textbf a|}\text{comp}_\textbf a \textbf b = \dfrac 4{13}\lang-5,12\rang = \left\lang\dfrac{-20}{13}, \dfrac{48}{13} \right\rang\ \blacksquare$ + +### Exercício 63 + +A *Lei do Paralelogramo* afirma que + +$$ +|\textbf a + \textbf b|^2 + |\textbf a - \textbf b|^2 = +2 |\textbf a|^2 + 2|\textbf b|^2 +$$ + +Dê uma interpretação geométrica da Lei do Paralelogramo e a demonstre + +#### Resolução + + + +Considere o paralelogramo acima. É possível aferir o comprimento de suas diagonais à partir do comprimento de seus lados. De fato, podemos aferi-las separadamente usando a *Lei dos Cossenos*: + +$|\overline{BD}|^2 = +|\overline{AD}|^2 + |\overline{AB}|^2 - 2 |\overline{AD}||\overline{AB}|\cos \theta +$ + +e + +$|\overline{AC}|^2 = +|\overline{AD}|^2 + |\overline{CD}|^2 - 2 |\overline{AD}||\overline{CD}|\cos(\pi - \theta) += \\ |\overline{AD}|^2 + |\overline{CD}|^2 - 2 |\overline{AD}||\overline{CD}|[\cos \pi \cdot \cos \theta + \cancel{\sin \pi \cdot \sin \theta}] = \\ +|\overline{AD}|^2 + |\overline{CD}|^2 + 2 |\overline{AD}||\overline{CD}|\cos\theta$ + +Substituindo o comprimento dos lados e diagonais por sua representação vetorial ($|\textbf a| = |\overline{AD}| = |\overline{BC}|$, $|\textbf b| = |\overline{AB}| = |\overline{CD}|$, $|\textbf{a + b}| = |\overline{AC}|$, $|\textbf a - \textbf b| = |\overline{BD}|$) e somando-se as equações anteriores, temos demonstrada a *Lei do Paralelogramo*: + +$$ ++ \begin{cases} +|\textbf a - \textbf b|^2 = +|\textbf a|^2 + |\textbf b|^2 +- 2 |\textbf a||\textbf b|\cos \theta \\ +|\textbf{a + b}|^2 = +|\textbf a|^2 + |\textbf b|^2 ++ 2 |\textbf a||\textbf b|\cos \theta \\ +\end{cases}\\\ \\ \therefore |\textbf a + \textbf b|^2 + |\textbf a - \textbf b|^2 = +2 |\textbf a|^2 + 2|\textbf b|^2\ \blacksquare +$$ + +## Capítulo 12.4 + +### Exercício 37 + +Utilize o produto misto para mostrar que os vetores $\textbf u = \textbf i + 5\textbf j - 2 \textbf k$, $\textbf v = 3 \textbf i - \textbf j$ e $\textbf w = 5 \textbf i + 9 \textbf j - 4 \textbf k$ são coplanares. + +#### Resolução + +Conforme a definição de produto misto, dados vetores são complanares se o produto misto destes for igual à 0. Avaliemos o presente caso. + +$ \textbf u (\textbf v \times \textbf w) = +\left|\begin{matrix} +1 & \phantom{-}5 & -2 \\ 3 & -1 & \phantom{-}0 \\ 5 & \phantom{-}9 & -4 +\end{matrix}\right| = +1 \left |\begin{matrix} +-1 & \phantom{-}0 \\ \phantom{-}9 & -4 +\end{matrix}\right | - 5 \left |\begin{matrix} +3 & \phantom{-}0 \\ 5 & -4 +\end{matrix}\right | + +(-2) \left |\begin{matrix} +3 & -1 \\ 5 & \phantom{-}9 +\end{matrix}\right | = \\\ \\ +4 - 5(-12) - 2(27 + 5) = 0\ \blacksquare $ + +### Exercício 49 + +Demonstre que $(\textbf a - \textbf b) \times (\textbf a + \textbf b) = 2(\textbf a \times \textbf b)$. + +#### Resolução + +Lembremos as seguintes propriedades: + +**P1.** $|\textbf a \times \textbf b| = |\textbf a||\textbf b| \sin \theta$ onde $\theta$ é o ângulo entre **a** e **b**, $0 \le \theta \le \pi$; + +**P2.** $\textbf a \times \textbf b = - \textbf b \times \textbf a$; + +**P3.** $\textbf a \times (\textbf b + \textbf c) = \textbf a \times \textbf b + \textbf a \times 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+$$ +|AB|^2 = |OA|^2 + |OB|^2 - 2 |OA||OB| \cos \theta +$$ + +Onde $|OA| = |\textbf a|$, $|OB| = |\textbf b|$ e $|AB| = |\textbf a - \textbf b|$. Ou seja, + +$|\textbf a - \textbf b|^2 = |\textbf a|^2 + |\textbf b|^2 - 2|\textbf a||\textbf b| \cos \theta = \\ +|\textbf a|^2 - 2 \textbf a \cdot \textbf b + |\textbf b|^2 = |\textbf a|^2 + |\textbf b|^2 - 2|\textbf a||\textbf b| \cos \theta= \\ - 2 \textbf a \cdot \textbf b = - 2|\textbf a||\textbf b| \cos \theta = \\ \textbf a \cdot \textbf b = |\textbf a||\textbf b| \cos \theta$ + +### Exemplo + +Determine o ângulo entre dois vetores $\textbf a = \lang 2, 2, -1\rang$ e $\textbf b \lang 5, -3, 2 \rang$. + +$\textbf a \cdot \textbf b = |\textbf a| |\textbf b| \cos \theta \implies \cos \theta = \dfrac{\textbf a \cdot \textbf b}{|\textbf a| |\textbf b|} = \dfrac{2(5) + 2(-3) + 2(-1)}{\sqrt{2^2 + 2^2 + (-1)^2} \cdot \sqrt{5^2 + (-3)^2 + 2^2}} = \dfrac 2{3\sqrt{38}}$ + +### Casos específicos + +Dois vetores $\textbf a$ e $\textbf b$ formam um ângulo + +- ortogonal se $\textbf a \cdot \textbf b = 0 \implies \theta = \frac 12 \pi$; + +- agudo se $\textbf a \cdot \textbf b > 0$; + +- obtuso se $\textbf a \cdot \textbf b < 0$. + +## Ângulos Diretores + +Os ângulos $\alpha$, $\beta$ e $\gamma$ (no intervalo $[0, \pi]$) que $\textbf a$ faz com os eixos coordenados positivos $x$, $y$ e $z$. + + + +Os cossenos desses ângulos diretores são chamados **cossenos diretores** do vetor $\textbf a$. + +$$ +\cos \alpha = \dfrac{a_1}{|\textbf a|};\, +\cos \beta = \dfrac{a_2}{|\textbf a|};\, +\cos \gamma = \dfrac{a_3}{|\textbf a|}. +$$ + +Onde + +$$ +\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 +$$ + +Por isso + +$$ +\textbf a = \lang a_1, a_2, a_3 \rang = +\lang|\textbf a| \cos \alpha,|\textbf a| \cos \beta, |\textbf a| \cos \gamma\rang = +|\textbf a|\lang \cos \alpha, \cos \beta, \cos \gamma\rang +$$ + +Disso implica que + +$$ +\dfrac 1{|\textbf a|} \textbf a = \lang \cos \alpha, \cos \beta, \cos \gamma\rang +$$ + +## Projeções + + + +A figura acima mostra as representações $\overrightarrow{PQ}$ e $\overrightarrow{PR}$ de dois vetores $\textbf a$ e $\textbf b$ com a mesma origem $P$. Se $S$ é o pé do perpendicular a partir de $R$ à reta contendo $\overrightarrow{PQ}$ , então o vetor coam representação $\overrightarrow{PS}$ é chamado **vetor projeção** de $\textbf b$ sobre $\textbf a$ e é denotado por $\text{proj}_\textbf a \textbf b$. + +$$ +\text{proj}_\textbf a \textbf b = +\left(\frac{\textbf a \cdot \textbf b}{|\textbf a|}\right) \frac{\textbf a }{|\textbf a|} +$$ + +> Onde $\frac{\textbf a }{|\textbf a|}$ é o *versor* (vetor unitário) de $\textbf a$. + +A projeção escalar de $\textbf b$ sobre $\textbf a$ (também chamada componente de $\textbf b$ ao longo de $\textbf a$) $\text{comp}_\textbf a \textbf b$ é definida como o módulo com sinal do vetor projeção, cujo valor é dado pelo número $|\textbf b| \cos \theta$, onde $\theta$ é o ângulo entre $\textbf a$ e $\textbf b$. + + + +$$ +\text{comp}_\textbf a \textbf b = |\textbf b| \cos \theta = +\frac{\textbf a \cdot \textbf b}{|\textbf a|} +$$ + +> Observe que o vetor projeção é a projeção escalar vezes o versor de $\textbf a$: +> +> $$ +> \text{proj}_\textbf a \textbf b = \frac{\textbf a }{|\textbf a|}\text{comp}_\textbf a \textbf b +> $$ \ No newline at end of file diff --git a/Cálculo II/Atividade 6/Produto Vetorial.md b/Cálculo II/Atividade 6/Produto Vetorial.md new file mode 100644 index 0000000..0af1c47 --- /dev/null +++ b/Cálculo II/Atividade 6/Produto Vetorial.md @@ -0,0 +1,105 @@ +# O Produto Vetorial + +## Definição + +Se $\textbf a = \lang a_1, a_2, a_3 \rang$ e $\textbf b = \lang b_1, b_2, b_3 \rang$, então **produto vetorial** ou **cruzado** de **a** e **b** é o vetor **c** perpendicular tanto à **a** e **b** descrito por + +$$ +\textbf c = \textbf a \times \textbf b = +\lang a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1 \rang +$$ + +> **Obs:** Definição de produto vetorial para vetores *tridimensionais*. + +Por ser ortogonal tanto à **a** e **b**, tem-se que: + +$$ +\textbf c \cdot \textbf a = \textbf c \cdot \textbf b = 0 += (\textbf a \times \textbf b) \cdot \textbf b += (\textbf a \times \textbf b) \cdot \textbf a +$$ + + + +> A regra da mão direita fornece a direção de $\textbf a \times \textbf b$, ortogonal ao plano que contêm **a** e **b**. + +### Propriedades + +Se **a**, **b** e **c** são vetores e *c* é um escalar, então + +1. $\textbf a \times \textbf a = 0$ + +2. $\textbf a \times \textbf b = - \textbf b \times \textbf a$ + +3. $(c \textbf a) \times \textbf b = c(\textbf a \times \textbf b) = \textbf a \times (c\textbf b)$ + +4. $\textbf a \times (\textbf b + \textbf c) = \textbf a \times \textbf b + \textbf a \times \textbf c$ + +5. $(\textbf a + \textbf b) \times \textbf c = \textbf a \times \textbf c + \textbf b \times \textbf c$ + +6. $\textbf a (\textbf b \times \textbf c) = (\textbf a \times \textbf b) \cdot \textbf c$ + +7. $\textbf a \times (\textbf b \times \textbf c) = + (\textbf a \cdot \textbf c)\textbf b - (\textbf a \cdot \textbf b)\textbf c$ + +### Exemplo + +Encontre um vetor perpendicular ao plano que passa pelos pontos + +$$ +P(1, 4, 6), Q(-2, 5, -1), R(1, -1, 1) +$$ + +#### Resolução + +O vetor $\overrightarrow{PQ} \times \overrightarrow{PR}$ é perpendicular a ambos $\overrightarrow{PQ}$ e $\overrightarrow{PR}$ e, portanto, perpendicular ao plano que passa por $P$, $Q$ e $R$. Tem-se que: + +$\overrightarrow{PQ} = (-2 - 1)\textbf i + (5 - 4)\textbf j + (-1 - 6)\textbf k = - 3\textbf i + \textbf j - 7\textbf k \\ \overrightarrow{PR} = (1 - 1)\textbf i + (-1 -4)\textbf j + (1 - 6)\textbf k = -5 \textbf j - 5 \textbf k \\ +\overrightarrow{PQ} \times \overrightarrow{PR} = \lang 1(-5) + 7(-5), -7(0) + 3(-5), -3(-5) - 1(0) \rang\\ = \lang-40,-15,15\rang = -5\lang 8,3,-3 \rang$ + +Assim, temos que $\lang-40,-15,15\rang$ é perpendicular ao plano e, no mais, todo múltiplo não nulo de $\lang 8,3,-3 \rang$ também o é. + +## Teorema + +Se $\theta$ é o ângulo entre **a** e **b**, $0 \le \theta \le \pi$, então + +$$ +|\textbf a \times \textbf b| = |\textbf a||\textbf b| \sin \theta = A +$$ + +Onde $A$ é a área descrita pelo paralelogramo formado entre os vetores. Assim, dois vetores são paralelos entre si se $\theta = k\pi$, $k \in \Z$. + + + +### Caso específico + +A ideia de produto vetorial aparece muito frequentemente em física. Por exemplo, ao apertarmos um parafuso aplicando uma força a uma chave de boca iremos girar o parafuso). O torque $\tau$ (em relação à origem) é definido +como sendo o produto cruzado dos vetores posição e força: + +$$ +\tau = \textbf r \times \textbf F +$$ + + + +Posto em termos da definição de produto vetorial, isso seria equivalente à + +$$ +|\tau| = |\textbf r \times \textbf F| = |\textbf r||\textbf F| \sin \theta +$$ + +## Produtos Triplos + +O produto $\textbf a \cdot (\textbf b \times \textbf c)$ que ocorre na Propriedade 5 da definição de produto vetorial é chamado **produto misto ou produto triplo escalar** dos vetores **a**, **b** e **c**. O significado geométrico do produto misto pode ser visto considerando-se o paralelepípedo determinado pelos vetores **a**, **b** e **c**. + + + +Assim sendo, o volume do paralelepípedo determinado pelos vetores a, b e c é o módulo do produto misto: + +$$ +V = | \textbf a \cdot (\textbf b \times \textbf c)| +$$ + +### Caso específico + +Se usarmos a fórmula anterior e descobrirmos que o volume do paralelepípedo determinado por **a**, **b** e **c** é 0, então os três vetores precisam pertencer ao mesmo plano; ou seja eles são **coplanares**. \ No newline at end of file diff --git a/Cálculo II/Atividade 7/Atividade 7.md b/Cálculo II/Atividade 7/Atividade 7.md new file mode 100644 index 0000000..4acd854 --- /dev/null +++ b/Cálculo II/Atividade 7/Atividade 7.md @@ -0,0 +1,60 @@ +# Atividade 7 + +> Resolução dos exercícios obrigatórios, feita por Guilherme de Abreu Barreto[^1]. + +## Capítulo 12.5 + +### Exercício 41 + +Use as intersecções com os eixos coordenados como uma ajuda para esboçar o plano $2x + 5y + z = 10$. + +#### Resolução + +![](/home/user/Public/USP/Sistemas%20de%20Informação/2º%20semestre/Cálculo%20II/Atividade%207/Imagens/2021-11-02-19-14-54-image.png) + +O plano cruza + +- o eixo x em $2x + 5(0) + (0) = 10 \implies x = 5$ + +- o eixo y em $2(0) + 5y + (0) = 10 \implies y = 2$ + +- o eixo z em $2(0) + 5(0) + z = 10 \implies z = 10$ + +### Exercício 71 + +Determine a distância do ponto $(1, -2, 4)$ ao plano $3x + 2y + 6z = 5$. + +#### Resolução + +$$ +D = \frac{|ax + by + cz + d|}{\sqrt{a^2 + b^2 + c^2}} += \frac{|3(1) + 2(-2) + 6(4) - 5|}{\sqrt{9 + 4 + 36}} = \frac{18}7 +$$ + +## Capítulo 14.1 + +### Exercício 47 + +Faça o mapa de contorno da função mostrando várias curvas de nível para + +$$ +f (x, y) = ye^x +$$ + +#### Resolução + +Mapa de contorno para valores $10 | k$ tais que $-10^6 \le k \le 10^6$: + +![](/home/user/Public/USP/Sistemas%20de%20Informação/2º%20semestre/Cálculo%20II/Atividade%207/Imagens/2021-11-02-21-15-01-image.png) + +### Exercício 71 + +Utilize um computador para traçar o gráfico da função $f(x, y) = 3x - x^4 - 4y^2 - 10xy$ usando vários domínios e pontos de vista. Imprima aquela que apre sente melhor os “picos e vales”. Você acha que essa função tem um valor máximo? Você poderia identificar os pontos do gráfico corres- pondentes aos “máximos locais”? E aos “mínimos locais”? + +#### Resolução + +![](/home/user/Public/USP/Sistemas%20de%20Informação/2º%20semestre/Cálculo%20II/Atividade%207/Imagens/2021-11-02-21-29-18-image.png) + +O gráfico acima corresponde a função descrita. Neste, é possível identificar dois máximos locais, para $z = 15$ e $z = 5$ e um mínimo local em $z = 0$. + +[^1]: nUSP 12543033; Turma 04 \ No newline at end of file diff --git a/Cálculo II/Atividade 7/Equações de Retas e Planos.md b/Cálculo II/Atividade 7/Equações de Retas e Planos.md new file mode 100644 index 0000000..9b54907 --- /dev/null +++ b/Cálculo II/Atividade 7/Equações de Retas e Planos.md @@ -0,0 +1,204 @@ +# Equações de Retas e Planos + +## Equação vetorial no espaço tridimensional + + + +$$ +\textbf r = \textbf r_0 + t\textbf v +$$ + +Onde: + +- $\textbf r_0 = \lang x_0, y_0, z_0 \rang$ é o vetor que parte da origem do sistemas de coordenadas $O$ e coincide com a origem do vetor $\textbf a$, $P_0$; + +- $\textbf r = \lang x, y, z \rang$ é o vetor que parte da origem do sistema de coordenadas $O$ e coincide com a extremidade oposta do vetor $\textbf a$, $P$; + +- $\textbf v = \lang a, b, c \rang$ é um vetor paralelo à $\textbf a$ partindo da origem do sistema de coordenadas, denominado **vetor diretor**; + +- $t$ é o escalar que multiplica $\textbf v$ de tal forma que este assume a mesma magnitude e sentido que $\textbf a$. + + > Assim, para diferentes valores de $t$ correspondem distintos pontos $P_0$ em $L$: + > + > + +Explicitando os componentes na fórmula anterior, tem-se: + +$$ +\lang x, y, z \rang = \lang x_0, y_0, z_0 \rang + \lang ta, tb, tc \rang += \lang x_0 + ta, y_0 + tb, z_0 + tc \rang +$$ + +Desta equação derivamos as seguintes **equações paramétricas**: + +$$ +\begin{cases}x = x_0 + at \\ y = y_0 + bt \\ z = z_0 + ct\end{cases} +$$ + +Podemos observar que entre estas $t$ é um fator comum. Logo, para qualquer vetor $\textbf a$ na reta $L$ em que $a,b,c \in \R^*$a seguinte igualdade é verdadeira: + +$$ +\frac{x - x_0}a = \frac{x - y_0}b = \frac{z - z_0}c +$$ + +Este conjunto de equações são denominadas **equações simétricas** de $L$. + +## Planos + +Um plano no espaço fica determinado se conhecermos um ponto $P_0(x_0, y_0, z_0)$ no plano e um vetor $\textbf n$, denominado **vetor normal**, ortogonal ao plano. + + + +Assim, seja $P(x, y, z)$ um ponto qualquer contido no plano tem-se que o vetor que liga $P_0$ à $P$ é $\textbf r - \textbf r_0$ tal que o produto vetorial $\textbf n \cdot (\textbf r - \textbf r_0) = 0$. Ou seja + +$$ +\textbf n \cdot \textbf r = \textbf n \cdot \textbf r_0 +$$ + +Escrito de maneira a explicitar os componentes dos vetores $\textbf n = \lang a,b,c \rang$, $\textbf r = \lang x,y,z \rang$ e $\textbf r_0 = \lang x_0,y_0,z_0 \rang$ temos + +$$ +\lang a,b,c \rang \cdot \lang x - x_0, y - y_0, z - z_0 \rang = +a(x - x_0) + b(y - y_0) + c(z - z_0) = 0 +$$ + +a **equação escalar do plano** que passa por $P_0$ com vetor normal $\textbf n$. + +No mais, a equação anterior pode ser simplificada como: + +$$ +ax + by + cz + d = 0 +$$ + +onde $d = -(ax_0 + by_0 + cz_0)$. Fórmula essa conhecida como **equação linear** em $x$, $y$, $z$. Uma importante aplicação desta equação é o cálculo da distância $D$ de um ponto com relação a um plano. Seja $x, y, z$ as coordenadas deste ponto, tem-se: + +$$ +D = \frac{|ax + by + cz + d|}{\sqrt{a^2 + b^2 + c^2}} +$$ + +### Equações simétricas na representação de planos + +Podemos pensar na reta como a intersecção de dois planos: + +$$ +\begin{matrix} \dfrac{x - x_0}a = \dfrac{x - y_0}b & e &\dfrac{x - y_0}b = \dfrac{z - z_0}c \end{matrix} +$$ + +Por exemplo, para uma reta $L$ descrita por + +$$ +\begin{matrix} \dfrac{x - 1}5 += \dfrac y{-2} & e & \dfrac y{-2} = \dfrac z{-3} \end{matrix} +$$ + +Tem-se o seguinte gráfico: + + + +## Exemplos + +### Exemplo 1 + +Mostre que as retas $L_1$ e $L_2$ com equações paramétricas dadas por + +$$ +\begin{cases} \begin{matrix} +x_1 = 1 + t & y_1 = -2 + 3t & z_1 = 4 - t \\ +x_2 = 2s & y_2 = 3 + s & z_2 = -3 + 4s +\end{matrix} \end{cases} +$$ + +são retas **reversas**, isto é, são retas que não se interceptam e não são paralelas (não pertencendo, portanto, a um mesmo plano). + +#### Resolução + +As retas não são paralelas, pois os componentes de seus vetores diretores $\lang 1, 3, 1 \rang$ e $\lang 2, 1, 4 \rang$ não são proporcionais entre si. + +As retas também não se intersectam pois, se houvesse intersecção, o seguinte sistema haveria uma solução: + +$$ +\begin{cases}1 + t = 2s \\ -2 + 3t = 3 + s \\ 4- t = -3 + 4s\end{cases} +$$ + +O que não é o caso para qualquer valor de $t$ e $s$. + +### Exemplo 2 + +Encontre uma equação do plano que passa pelos pontos $P(1, 3, 2)$, $Q(3, -1, 6)$ e $R(5, 2, 0)$. + +#### Resolução + +Os vetores **a** e **b** correspondentes a $\overrightarrow{PQ}$ e $\overrightarrow{PR}$ são + +$$ +\begin{cases} +\textbf a = \lang 3 - 1, -1 - 3, 6 - 2 \rang = \lang 2, -4, 4 \rang \\ +\textbf b = \lang 5 - 1, 2 - 3, 0 - 2 \rang = \lang 5, 2, 0 \rang +\end{cases} +$$ + +Como tanto **a** quanto **b** pertencem ao plano, o produto vetorial $\textbf a \times \textbf b$ é ortogonal ao plano e pode ser tomado como o vetor normal **n**. Assim, + +$$ +\textbf n = \textbf a \times \textbf b = \left|\begin{matrix} +\textbf i & \textbf j & \textbf k \\ +2 & -4 & 4 \\ +4 & -1 & -2 +\end{matrix}\right| = \left|\begin{matrix} +-4 & 4 \\ +-1 & -2 +\end{matrix}\right| \textbf i - \left|\begin{matrix} +2 & 4 \\ +4 & -2 \end{matrix}\right| \textbf j + \left|\begin{matrix} +2 & -4 \\ +4 & -1 +\end{matrix}\right| \textbf k =\\ +(8 + 4) \textbf i - (-4 - 16) \textbf j + (-2 + 16) \textbf k = +\lang 12, 20, 14 \rang \equiv \lang 6, 10, 7 \rang +$$ + +Com o ponto $P(1, 3, 2)$ e o vetor normal **n**, uma equação do plano é + +$$ +6(x - 1) + 10(y - 3) + 7(z - 2) = 0 \implies 6x + 10y + 7z = 50 +$$ + +### Exemplo 3 + +**a.** Determine o ângulo entre os planos $x + y + z = 1$ e $x - 2y + 3z = 1$ + +Os vetores normais a esses planos são + +$$ +\begin{matrix} +\textbf n_1 = \lang 1, 1, 1 +\rang & \textbf n_2 = \lang 1, -2, 3\rang +\end{matrix} +$$ + +Portanto, se $\theta$ é o ângulo entre os dois planos, + +$$ +\cos \theta = \frac{\textbf n_1 \cdot \textbf n_2}{|\textbf n_1||\textbf n_2|} += \frac{1(1) + 1(-2) + 1(3)}{\sqrt{1 + 1 + 1} \sqrt{1 + 4 + 9}} = +\frac 2{\sqrt{42}}\\\ \\ +\therefore \theta = \cos^{-1} \left(\frac 2{\sqrt{42}}\right) \approx 72\degree +$$ + +**b.** Primeiro precisamos encontrar um ponto em $L$. Por exemplo, podemos achar o ponto onde a reta intercepta o plano $xy$ tomando $z = 0$ na equação dos dois planos. Isso fornece as equações $x + y = 1$ e $x - 2y = 1$, cuja solução é $x = 1$, $y = 0$. Portanto, o ponto $(1, 0, 0)$ encontra-se em $L$. + +Observe que, como $L$ pertence a ambos os planos, é perpendicular ao vetor normal de ambos os planos. Então, um vetor **v** paralelo a $L$ é dado pelo produto vetorial + +$$ +\textbf v = \textbf n_1 \times \textbf n_2 = \left|\begin{matrix} +\textbf i & \textbf j & \textbf k \\ +1 & 1 & 1 \\ +1 & -2 & 3 +\end{matrix} \right| = 5 \textbf i - 2 \textbf j - 3 \textbf k +$$ + +e assim as equações simétricas de $L$ podem ser escritas como + +$$ +\frac{x - 1}5 = \frac y{-2} = \frac z{-3} +$$ \ No newline at end of file diff --git a/Cálculo II/Atividade 7/Funções de Múltiplas Variáveis.md b/Cálculo II/Atividade 7/Funções de Múltiplas Variáveis.md new file mode 100644 index 0000000..a12da0f --- /dev/null +++ b/Cálculo II/Atividade 7/Funções de Múltiplas Variáveis.md @@ -0,0 +1,23 @@ +# Funções de Múltiplas Variáveis + +## Funções de Duas Variáveis + +Uma função $f$ de duas variáveis é uma regra que associa a cada par ordenado de números reais $(x, y)$ de um conjunto $D$ um único valor real, denotado por +$f (x, y)$. O conjunto $D$ é o domínio de $f$ e sua imagem é o conjunto de valores possíveis de $f$, ou seja, $f= \{(x, y): x, y \in D\}$. Por exemplo, funções da forma $f: \R^2 \to \R$. + +### Gráficos + +Se $f$ é uma função de duas variáveis com domínio $D$, então o gráfico de $f$ é +o conjunto de todos os pontos $(x, y, z)$ em $\R^3$ tal que $z = f (x, y)$ e $(x, y)$ pertença a $D$. + + + +#### Curvas de nível + +As curvas de nível de uma função $f$ de duas variáveis são aquelas com equação $f (x, y) = k$, onde $k$ é uma constante (na imagem de $f$). + +![](/home/user/Public/USP/Sistemas%20de%20Informação/2º%20semestre/Cálculo%20II/Atividade%207/Imagens/2021-11-02-20-15-15-image.png) + +## Funções de Três ou Mais Variáveis + +Uma função com três variáveis $f$, é uma regra que associa a cada tripla ordenada $(x, y, z)$ em um domínio $D \subset \R^3$ um único número real, denotado por $f(x, y, z)$. Enquanto uma função com $n$ variáveis é uma regra que associa um número $z = f(x_1, x_2, \dots, x_n)$ a uma n-upla $(x_1, x_2, \dots, x_n)$ de números reais. 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Turma 04 \ No newline at end of file diff --git a/Cálculo II/Atividade 8/Derivadas parciais.md b/Cálculo II/Atividade 8/Derivadas parciais.md new file mode 100644 index 0000000..7d63c3d --- /dev/null +++ b/Cálculo II/Atividade 8/Derivadas parciais.md @@ -0,0 +1,41 @@ +# Derivadas Parciais + +Se $f$ é uma função de duas variáveis ($x$, $y$), suas derivadas parciais são as funções $f_x$ e $f_y$ definidas por + +$$ +f_x (x, y) = \lim_{h \to 0} \frac{f[(x + h),y] - f(x,y)}h \\\ \\ +f_y (x, y) = \lim_{h \to 0} \frac{f[x,(y + h)] - f(x,y)}h +$$ + +Outras notações utilizadas para derivadas parciais: + +$$ +f_x(x,y) = f_x = \frac{\partial f}{\partial x} += \frac{\partial}{\partial x} f(x,y) = \frac{\partial z}{\partial x} = +f_1=D_1f = D_xf +$$ + +Onde $z = f(x,y)$ e o numerador $1$ é um índice que indica a primeira variável do par ordenado. + +## Regra para determinar Derivadas Parciais de $\mathbf{z = f(x,y)}$ + +1. Para determinar $f_x$, trate $y$ como uma constante e derive $f(x,y)$ com relação a $x$. +2. Para determinar $f_y$, trate $x$ como uma constante e derive $f(x,y)$ com relação a $y$. + +## Derivadas de Ordem Superior + +Se $f$ é uma função de duas variáveis, suas derivadas parciais $f_x$ e $f_y$ são funções de duas variáveis, de modo que podemos considerar novamente suas derivadas parciais$ (f_x)_x$, $(f_x)_y$, $(f_y)_x$ e $(f_y)_y$, chamadas **derivadas parciais de segunda ordem** de $f$. Se $z = f(x, y)$, usamos a seguinte notação, por exemplo: + +$$ +(f_x)_x = f_{xx} = f_{11} += \frac \partial{\partial x} \left( \frac {\partial f}{\partial x}\right) += \frac {\partial^2f}{\partial x^2} = \frac {\partial^2z}{\partial x^2} +$$ + +### Teorema de Clairaut + +Suponha que $f$ seja definida em uma bola aberta $D$ que contenha o ponto $(a, b)$. Se as funções $f_{xy}$ e $f_{yx}$ forem ambas contínuas em D, então + +$$ +f_{xy}(a,b) = f_{yx}(a,b) +$$ \ No newline at end of file diff --git a/Cálculo II/Atividade 8/Limites e continuidade.md b/Cálculo II/Atividade 8/Limites e continuidade.md new file mode 100644 index 0000000..72e9f9a --- /dev/null +++ b/Cálculo II/Atividade 8/Limites e continuidade.md @@ -0,0 +1,13 @@ +# Limites e continuidade + +Seja $f$ uma função de duas variáveis cujo domínio $D$ contém pontos arbitrariamente próximos de $(a,b)$. Dizemos que o limite de $f(x,y)$ quando $(x,y)$ tende a $(a,b)$ é $L$ e escrevemos + +$$ +\lim_{(x,y) \to (a,b)} f(x,y) = L +$$ + +se para todo número $\varepsilon > 0$ houver um número correspondente de $\delta > 0$ tal que se $(x,y) \in D$ e $0 < \sqrt{(x - a)^2 + (y - b)^2} < \delta$ então $|f(x,y) - L| < \varepsilon$[^1]. + +Se $f (x, y) \to L_1$ quando $(x, y) \to (a, b)$ ao longo do caminho $C_1$ e $f (x, y) \to L_2$ quando $(x, y) \to (a, b)$ ao longo do caminho $C_2$, com $L_1 \not = L_2$, então $\lim_{(x, y) \to (a, b)} f (x, y)$ não existe. + +[^1]: Observe que $|f(x,y) - L |$ corresponde à distância entre os números $f (x, y)$ e $L$, e $\sqrt{(x - a)^2 + (y - b)^2}$ é a distância entre o ponto $(x, y)$ e o ponto $(a, b)$. Noutras palavras, a definição diz que a distância entre $f (x, y)$ e $L$ pode ser feita arbitrariamente pequena se tornarmos a distância de $(x, y)$ a $(a, b)$ suficientemente pequena (mas não nula). \ No newline at end of file diff --git a/Cálculo II/Atividade 9/Atividade 9.md b/Cálculo II/Atividade 9/Atividade 9.md new file mode 100644 index 0000000..682ffe5 --- /dev/null +++ b/Cálculo II/Atividade 9/Atividade 9.md @@ -0,0 +1,123 @@ +# Atividade 9 + +> Resolução dos exercícios obrigatórios, feita por Guilherme de Abreu Barreto[^1]. + +## Capítulo 14.4 + +### Exercício 24 + +O índice de sensação térmica $W$ é a temperatura sentida quando a temperatura real é $T$ e a velocidade do vento, $v$. Portanto, podemos escrever $W = f (T, v)$. A tabela de valores a seguir foi extraída da Tabela 1 da Seção 14.1. Use essa tabela para determinara aproximação linear da função de sensação térmica quando $T$ estiver a -15 ºC e $v$ estiver próximo de 50 km/h. Estime, a seguir, a sensação térmica quando a temperatura estiver a -17 ºC +e a velocidade do vento for de 55 km/h. + + + +#### Resolução + +A aproximação linear para $f(T,v)$ é dada por: + +$$ +f(T,v) \approx f(a,b) + f_T(a,b)(T - a) + f_v(a,b)(v - b) +$$ + +Para valores $a \approx T$ e $b \approx v$. Assim sendo, para estimarmos $f(-17,55)$ utilizaremos o valor descrito na tabela para $f(a,b) = f(-15,60)$ e aqueles adjacentes a este. Assim, temos que + +$f_T(-15, 60) \approx \displaystyle \lim_{h \to 5} \dfrac{f(-15 + h, 60) - f(-15,60)}{2h}\\\ \\ + \lim_{h \to -5} \dfrac{f(-15 +h, 60) - f(-15,60)}{2h} = \dfrac{\dfrac{-23 + 30}5 + \dfrac{-36 + 30}{-5}}2 = \dfrac{13}{10}$ + +e + +$f_v(-15, 60) \approx \displaystyle \lim_{h \to 10} \dfrac{f(-15, 60 + h) - f(-15,60)}{2h}\\ \ + \lim_{h \to -10} \dfrac{f(-15, 60 + h) - f(-15,60)}{2h} = \dfrac{\dfrac{30 - 30}{10} + \dfrac{30 - 29}{-10}}2 = - \dfrac 1{20}$ + +Logo, + +$f(-17,55) \approx f(-15, 60) + f_T(-15,60)(-17 + 15) + f_v(-15,60)(55 - 60) \\\ \\ \approx -30 + \dfrac{13}{10}(-2) - \dfrac 1{20}(-5) \approx -32,25\ \blacksquare$ + +### Exercício 42 + +Suponha que você precise saber uma equação do plano tangente à superfície $S$ no ponto $P(2, 1, 3)$. Você não tem uma equação para $S$, mas sabe que as curvas + +- $\textbf r_1(t) = \lang 2 + 3t, 1 - t^2, 3 - 4t + t^2\rang$ + +- $\textbf r_2(u) = \lang 1 + u^2, 2u^3 - 1, 2u + 1 \rang$ + +ambas estão em $S$. Encontre uma equação para o plano tangente em $P$. + +#### Resolução + +Podemos deduzir onde as retas passam pelo ponto $P$ fazendo a seguinte comparação: + +- Se $\textbf r_1(t) = \lang 2 + 3t, 1 - t^2, 3 - 4t + t^2\rang = \lang 2, 1, 3 \rang$, então + $\begin{cases} + 2 + 3t = 2 \\ + 1 - t^2 = 1 \\ + 3 - 4t + t^2 = 3 + \end{cases} \therefore t = 0$ + Portanto, $\textbf r_1(t)$ cruza $P$ em $\textbf r_1(0)$. + +- Se $\textbf r_2(u) = \lang 1 + u^2, 2u^3 - 1, 2u + 1 \rang= \lang 2, 1, 3 \rang$, então + $\begin{cases} + 1 + u^2 = 2 \\ + 2u^3 - 1 = 1 \\ + 2u + 1 = 3 + \end{cases} \therefore u = 1$ + Portanto, $\textbf r_2(u)$ cruza $P$ em $\textbf r_2(1)$. + +Derivamos então as equações das curvas para obter a reta tangente destas: + +- $\textbf r_1(t) = \lang 2 + 3t, 1 - t^2, 3 - 4t + t^2\rang \implies \textbf r_1'(t) = \lang 3, - 2t, 2t - 4 \rang$ + +- $\textbf r_2(u) = \lang 1 + u^2, 2u^3 - 1, 2u + 1 \rang \implies \textbf r_2'(u) = \lang 2u, 6u, 2 \rang$ + +Com as retas tangentes conseguimos obter a reta normal $\textbf n$, perpendicular à ambas, no ponto $P = (2, 1, 3)$: + +$\textbf r_1'(0) \times \textbf r_2'(1) = \lang 3, 0 , -4 \rang \times \lang 2, 6, 2 \rang \\ = \lang 0 \cdot 2 - (-4 \cdot 6), -4 \cdot 2 - 3 \cdot 2, 3 \cdot 6 - 0 \cdot 2 \rang = \lang 24, -14, 18 \rang$ + +Por vez, a reta normal nos permite descrever a **equação linear** do plano sobre o ponto $P$: + +$24 x - 14y + 18z - (24 \cdot 2 - 14 \cdot 1 + 18 \cdot 3) = 0\\ \\ \implies 12 x - 7y + 9z - (12 \cdot 2 - 7 \cdot 1 + 9 \cdot 3) = 0\\ \\ \implies 12x - 7y + 9z - 44 = 0\ \blacksquare$ + +## Capítulo 14.5 + +### Exercício 43 + +Um lado de um triângulo está aumentando em uma taxa de $3\ cm/s$ e um segundo lado está decrescendo em uma taxa de $2\ cm/s$. Se a área do triângulo permanece constante, a que taxa varia o ângulo entre os lados quando o primeiro lado tem $20 cm$ de comprimento, o segundo lado tem $30 cm$ de comprimento e +o ângulo é $\frac \pi6$? + +#### Resolução + +Denominemos por $x$ o primeiro lado, $y$ o segundo e $\theta$ o ângulo entre eles. Pela aplicação da Lei dos senos, podemos aferir a área do triângulo $A$ como sendo $A = \frac{xy\sin\theta}2$ Assim, o valor de $A$ se dá em função de $x$, $y$ e $\theta$ e estes por vez se dão em função do tempo $t$. Sabemos pelo enunciado que a taxa de variação do comprimento de $x$, $\frac{dx}{dt} = 3$, e $y$, $\frac{dy}{dt} = -2$. Também, que para a área $A$ não ocorre variação, $\frac{dA}{dt} = 0$. Buscamos aqui saber a taxa de variação do ângulo $\theta$, $\frac{d\theta}{dt}$. Ora, podemos relacionar estes dados fazendo uso da Regra da Cadeia e inferi-la: + +$\displaystyle \frac{dA}{dt} = \frac{\partial A}{\partial x}\frac{dx}{dt} + \frac{\partial A}{\partial y}\frac{dy}{dt} + \frac{\partial A}{\partial \theta}\frac{d\theta}{dt} \implies 0 = \frac{3y \sin \theta}2 - \frac{2 x \sin \theta}2 + \dfrac{xy \cos \theta}2\frac{d\theta}{dt} \\\ \\ \implies \frac{d\theta}{dt} = \frac{2x\sin \theta - 3y\sin \theta}{xy\cos \theta} = \frac{\sin \theta}{\cos \theta} \cdot \frac{2x - 3y}{xy} = \tan \left(\frac \pi 6\right) \cdot \frac{2 \cdot 2\cancel 0 - 3 \cdot 3\cancel 0}{60\cancel 0}\\\ \\ = \frac{\sqrt 3}3 \cdot - \frac 1{12} = - \frac{\sqrt 3}{36}\ \blacksquare$ + +### Exercício 59 + +A Equação 6 é uma fórmula para a derivada $dy/dx$ de uma função definida implicitamente por uma equação $F (x, y) = 0$, sendo que $F$ é diferenciável e $F_y \not = 0$. Comprove que se $F$ tem derivadas contínuas de segunda ordem, então uma fórmula para a segunda derivada de $y$ é + +$$ +\frac{d^2y}{dx^2} = - \frac{F_{xx}F_y^2 - 2F_{xy}F_xF_y + F_yyF_x^2}{F_y^3} +$$ + +#### Resolução + +Dado que a função foi definida implicitamente da maneira descrita pelo enunciado, sabemos que $\frac{dy}{dx} = - \frac{F_x}{F_y}$. Denominemos $G(x,y) = - \frac{F_x}{F_y}$. Ao derivarmos ambos os lados da equação e utilizarmos a Regra da Cadeia, teremos: + +$$ +\displaystyle \frac{d^2y}{dx^2} = \frac{\partial G}{\partial x} \cdot \cancel{\frac{dx}{dx}}\ 1 + \frac{\partial G}{\partial y} \cdot \frac{dy}{dx} +$$ + +Sendo que + +- $\dfrac{\partial G}{\partial x} = \dfrac{\partial}{\partial x}\left(- \dfrac{F_x}{F_y}\right) = - \dfrac{F_yF_{xx} - F_xF_{yx}}{F_y^2}$ + +- $\dfrac{\partial G}{\partial y} = \dfrac{\partial}{\partial y}\left(- \dfrac{F_x}{F_y}\right) = - \dfrac{F_yF_{xy} - F_xF_{yy}}{F_y^2}$ + +Assim, + +$\displaystyle \frac{d2y}{dx2} = - \dfrac{F_yF_{xx} - F_xF_{yx}}{F_y^2} + \left(- \dfrac{F_yF_{xy} - F_xF_{yy}}{F_y^2}\right)\left(- \dfrac{F_x}{F_y}\right) = \\\ \\ - \dfrac{F_{xx}F_y^2 - F_{yx} F_xF_y - F_{xy} F_yF_x + F_{yy}F_x^2}{F_y^3}$ + +Consideremos agora que $F$ tem derivadas de segunda ordem contínuas então, pelo Teorema de Clauraut, $F_{xy} = F_{yx}$ e + +$$ +\frac{d^2y}{dx^2} = - \frac{F_{xx}F_y^2 - 2F_{xy}F_xF_y + F_yyF_x^2}{F_y^3} \ 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zGV=1)rN-}YLVva#LKQ#`7&$gjFXl?ck&u{pS5xJ;KbXf7X1zE+FR=b}YM!2f1s6IT z>4#!9Nc&{*zX^{rzO(511fKSHcdw9aFG>oA!(70Cg + +Ou seja para valores próximos de $a$, a curva descrita pela função $f(x)$ se aproxima de uma reta que contém o valor $a$. Desta forma, é possível utilizar uma função afim $L(x)$ de maneira a obter uma aproximação da função geral $f(x)$, tal que $L(x) = f(a) + f'(a)(x-a) \approx f(x)$. Onde $L(x)$ é denominada a **linearização** de $f$ no ponto $a$. + +#### Exemplo + +Calculemos o valor aproximado de $\sqrt[3]{25}$. + +##### Resolução + +1. Seja $f(x) = x^{\frac 13}$, o problema consiste em encontrar o valor de $f(25)$. + +2. Precisamos de um valor próximo de 25, do qual saibamos qual é a raiz cúbica. Sabemos que $f(27) = 3$, então usemos $a = 27$. + +3. Derivando $f(x)$ e encontrando o valor de $f'(a)$: + +$$ +f'(x) = \frac{x^{- \frac 23}}3 = \frac 1{3 \sqrt[3]{x^2}} \implies +f'(27) = \dfrac 1{3\sqrt[3]{27^2}} = \frac 1{27} +$$ + +4. Usando a aproximação linear: + +$$ +f(25) \approx f(27) + f'(27)(25 - 27) = 3 - \frac 2{27} \approx 2,926 +$$ + +5. O resultado 2,926 é um valor bem próximo, e portanto uma boa aproximação, do valor real 2,924. + +### Para funções com duas variáveis + +O mesmo procedimento pode ser realizado uma função com duas variáveis $f(x,y)$ fazendo uso de suas derivadas parciais de $x$ e $y$: $f_x$ e $f_y$. Tal que chegamos na seguinte fórmula: + +$$ +L(x,y) = f(a,b) + f_x(a,b)(x - a) + f_y(a,b)(y - b) \approx f(x,y) +$$ + +Onde $a$ e $b$ são constantes tais que $x \approx a$ e $y \approx b$. + +### Para funções com três ou mais variáveis + +De forma análoga, temos: + +$L(x,y,z) = f(a,b) + f_x(a,b,c)(x - a) + f_y(a,b,c)(y - b)\\ + f_z(a,b,c)(z - c) \approx f(x,y,z)$ + +E assim por diante. + +## Diferenciabilidade + +Se $z = f(x,y)$, então $f$ é **diferenciável** em $(a,b)$ de $\Delta z$ puder ser expresso na forma + +$$ +\Delta z = f_x(a,b)\Delta x + f_y(a,b)\Delta y + \varepsilon_1\Delta x + \varepsilon_2 \Delta y +$$ + +onde tanto $\varepsilon_1$ e $\varepsilon_2 \to 0$ quando $(\Delta x,\Delta y) \to (0,0)$ + +Ou seja, uma função é diferenciável se, e somente se, sua aproximação linear fornece uma boa aproximação para $f(x,y)$ para valores próximos de $f(a,b)$. + +### Teorema + +Se as derivadas parciais $f_x$ e $f_y$ existirem perto do ponto $(a, b)$ e forem contínuas em $(a, b)$, então $f$ é diferenciável em $(a, b)$. + +## Diferenciais + +### Para funções de uma única variável + +Para uma função de uma única variável, $y = f (x)$, definimos a diferencial $dx$ como uma variável independente; ou seja, $dx$ pode valer qualquer número real. A diferencial de $y$ é definida como + +$$ +dy = f'(x)\ dx +$$ + + + +### Para funções de duas variáveis + +Para uma função de duas variáveis, $z = f (x, y)$, definimos as diferenciais $dx$ e $dy$ como variáveis independentes; ou seja, podem ter qualquer valor. Então a diferencial $dz$ também chamada de **diferenciação total**, é definida por + +$$ +dz = f_x(x,y)\ dx + f_y(x,y)\ dy += \frac{\partial z}{\partial x}\ dx + \frac{\partial z}{\partial y}\ dy +$$ + +> Algumas vezes a notação utilizada para a diferenciação total é $df$. + +E assim, com a notação diferencial, a aproximação linear pode ser escrita como + +$$ +f(x,y) \approx f(a,b) + dz +$$ + + + +### Para funções de três variáveis + +Analogamente, + +$$ +dw = yz\ dx + xz\ dy + xy\ dz += \frac{\partial w}{\partial x}\ dx + \frac{\partial w}{\partial y}\ dy ++ \frac{\partial w}{\partial z}\ dz +$$ \ No newline at end of file diff --git a/Cálculo II/Atividade 9/Regra da Cadeia.md b/Cálculo II/Atividade 9/Regra da Cadeia.md new file mode 100644 index 0000000..8b523b7 --- /dev/null +++ b/Cálculo II/Atividade 9/Regra da Cadeia.md @@ -0,0 +1,76 @@ +# Regra da Cadeia + +Para as funções de mais de uma variável, a Regra da Cadeia tem muitas versões, cada uma delas fornecendo uma regra de derivação de uma função composta. + +## Caso 1 + +Suponha que $z = f (x,y)$ seja uma função diferenciável de $x$ e $y$, onde $x = g(t)$ e $y = h(t)$ são funções diferenciáveis de $t$. Então $z$ é uma função diferenciável de $t$ e + +$$ +\frac{dz}{dt} = \frac{\partial z}{\partial x} \frac{dx}{dt} + +\frac{\partial z}{\partial y} \frac{dy}{dt} +$$ + +## Caso 2 + +Suponha que $z = f(x,y)$ seja uma função diferenciável de $x$ e $y$, onde $x = g(s,t)$ e $y = h(s,t)$ são funções diferenciáveis de $s$ e $t$. Então + +$$ +\begin{matrix} +\displaystyle \frac{\partial z}{\partial s} = +\frac{\partial z}{\partial x} \frac{\partial x}{\partial s} + +\frac{\partial z}{\partial y} \frac{\partial y}{\partial s}& & +\displaystyle \frac{\partial z}{\partial t} = +\frac{\partial z}{\partial x} \frac{\partial x}{\partial t} + +\frac{\partial z}{\partial y} \frac{\partial y}{\partial t} +\end{matrix} +$$ + +Denomina-se $s$ e $t$ as variáveis **independentes**, $x$ e $y$ as variáveis **intermediárias**, e $z$ a variável **dependente**. Tal qual ilustra o seguinte **diagrama em árvore**: + + + +## A Regra da Cadeia (Versão Geral) + +Suponha que $u$ seja uma função diferenciável de $n$ variáveis $x_1, \dots , x_n$ onde cada $x$ é uma função diferenciável de $m$ variáveis $t_1, \dots , t_m$. Então $u$ é uma função de $t_1, \dots , t_m$ e + +$$ +\frac{\partial u}{\partial t_i} = +\sum^i_{j = 1} \frac{\partial u}{\partial x_j} +\frac{\partial x_j}{\partial t_i} +$$ + +Por exemplo, a Regra da Cadeia para o caso onde $w = f(x,y,z,t)$ e $x(u,v)$, $y = y(u,v)$, $z = z(u,v)$ e $t = t(u,v)$ é exemplificada pela seguinte diagrama de árvore + + + +e pode ser descrito em suas parciais por: + +$$ +\frac{\partial w}{\partial u} = +\frac{\partial w}{\partial x}\frac{\partial x}{\partial u} + +\frac{\partial w}{\partial y}\frac{\partial y}{\partial u} + +\frac{\partial w}{\partial z}\frac{\partial z}{\partial u} + +\frac{\partial w}{\partial t}\frac{\partial t}{\partial u} \\\ \\ +\frac{\partial w}{\partial v} = +\frac{\partial w}{\partial x}\frac{\partial x}{\partial v} + +\frac{\partial w}{\partial y}\frac{\partial y}{\partial v} + +\frac{\partial w}{\partial z}\frac{\partial z}{\partial v} + +\frac{\partial w}{\partial t}\frac{\partial t}{\partial v} +$$ + +## Diferenciação implícita + +Suponha que $z$ seja dado implicitamente como uma função $z = f(x,y)$ por uma equação da forma $F(x,y,z) = 0$. Isso significa que $F(x,y,f(x,y)) = 0$ para todo $(x,y)$ no domínio de $f$. Se $F$ e $f$ forem diferenciáveis, + +$$ +\begin{matrix} +\dfrac{\partial z}{\partial x} = +\dfrac{\dfrac{\partial F}{\partial x}}{\dfrac{\partial F}{\partial z}} +& & +\dfrac{\partial z}{\partial y} = +\dfrac{\dfrac{\partial F}{\partial y}}{\dfrac{\partial F}{\partial z}} +\end{matrix} +$$ + +$$ \ No newline at end of file diff --git a/Introdução à Análise de Algoritmos/Correção.md b/Introdução à Análise de Algoritmos/Correção.md deleted file mode 100644 index c7a34ea..0000000 --- a/Introdução à Análise de Algoritmos/Correção.md +++ /dev/null @@ -1,61 +0,0 @@ -# Correção - -- Identificar invariáveis - -- Inicialização - -- Manutenção - -- Término - -## Anotações GDB - -- Compilar um programa com a flag `-g` faz com que este programa armazene seu código fonte, permitindo-o ser debugado em sua execução. - -- Para se debugar um programa, chama-se este com o comando `gdb` no terminal. - -- `run` roda o programa. - -- `layout next` muda a forma de exibição da interface. - -- `refresh` atualiza a interface. - -- `next`, ou `n`, avança o programa em uma linha de comando *no escopo atual*. - -- `print`, ou apenas `p`, pode ser utilizado para realizar calculos aritméticos com o valor de variáveis. - -- `print var` apresenta o valor da variável var. - -- `print vector@x` apresenta os primeiros `x` valores de um vetor `vector`. - -- `print *vector@x` apresenta os valores apontados por um vetor de ponteiros `vector`. - -- `step` avança o programa em uma linha de comando *no escopo global*, entrando em funções, etc. - -- `break x` coloca um ponto de interrupção da execução na linha `x`. - -- `break function` coloca um ponto de interrupção da execução no início de uma função `function`. - -- `info break` lista todos os pontos de interrupção; - -- `del x` apaga o breakpoint de número `x`, tal qual descrito por `info break`. - -- `clear function` retira os pontos de interrupção colocados na função `function`. - -- `display i` para cada comando seguinte, mostrar o valor de `i`. `undisplay n` cessa este comportamento para a variável de índice `n`. - -- `watch [expressão]` (*watchpoint*) interrompe a execução do programa toda vez que o valor de uma expressão se altera. - -- `whatis i`, ou `what i`, mostra o tipo da variável `i`. - -- `backtrace` (`bt`) no caso de um Segmentation Fault, aponta-se a linha em que este ocorreu e os valores das variáveis à este ponto. - -- `finish` executa a função atual e interrompe a execução quando esta termina, indicando onde no código esta retorna e com qual valor. - -- `quit` sai do debugger. - -## Comandos avançados - -- `target record-full` grava a execução e permite voltar para passos anteriores com `reverse-next`, `reverse-step`, `reverse-continue` (`rn`, `rs`). - -- `set var i = x` altera o valor da variável `i` para `x` e permite a continuidade da execução do código. \ No newline at end of file diff --git a/Introdução à Análise de Algoritmos/Imagens/2021-10-17-19-22-33-image.png b/Introdução à Análise de 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zHYdy%vH&w%4~_wt0+RTaTCG8%738<@?mQM`sl@J;RX%u Feita por Guilherme de Abreu Barreto[^1] + +## Teoria dos Conjuntos + +### Exercício 1 + +**(a)** O conjunto dos planetas no Sistema Solar; + +**(b)** O conjunto dos Estados Federativos da República do Brasil. + +**(c)** O conjunto dos números naturais pares; + +**(d)** O conjunto de potências de 2 para qualquer expoente ${x \in \N : x \ge 1}$; + +**(e)** O conjunto dos números primos. + +### Exercício 2 + +$A \cap B \cap C$: o conjunto das argentinas residentes no Brasil; + +$B \backslash A$: o conjunto dos residentes no Brasil que não são argentinos; + +$C \backslash A$: o conjunto das mulheres no mundo que não são argentinas; + +$C \backslash B$: o conjunto das mulheres no mundo que não residem no Brasil; + +$B \backslash C$: o conjunto de residentes homens no Brasil. + +### Exercício 3 + +$$ +\begin{matrix} +\{\} & \{a\} & \{b\} & \{c\} \\ +\{d\} & \{a,b\} & \{a,c\} & \{a,d\} \\ +\{b,c\} & \{b,d\} & \{c,d\} & \{a,b,c\} \\ +\{a,b,d\} & \{a,c,d\} & \{b,c,d\} & \{a,b,c,d\} +\end{matrix} +$$ + +### Exercício 4 + +$$ +\binom 52 = \frac{5!}{2!(5 - 2)!} = 10 +$$ + +### Exercício 5 + +A relação de contingência é transitiva. Então se A está contido em B, e B está contido em C, A está contido em C. Tal qual ilustra a seguinte imagem (em ordem reversa): + + + +### Exercício 6 + +Se $A$ está contido em $B$, então a união de $A$ com $C$ está contida pela união de $B$ com $C$. De fato, o seguinte diagrama de Venn demonstra esta proposição: + +| A, B, C | B ∪ C | B ∪ C ⊆ A ∪ C | +|:------------------------------------------------------------------------------------------------------------------------------------------------------------:|:------------------------------------------------------------------------------------------------------------------------------------------------------------:|:------------------------------------------------------------------------------------------------------------------------------------------------------------:| +| ![](/home/user/Public/USP/Sistemas%20de%20Informação/2º%20semestre/Matemática%20Discreta%20I/Lista%203/Imagens/0666956f5b22720e81a8b19e0123d0bbb181e628.png) | ![](/home/user/Public/USP/Sistemas%20de%20Informação/2º%20semestre/Matemática%20Discreta%20I/Lista%203/Imagens/5bafa135117158e2f35d20cd8722390c445cb0db.png) | ![](/home/user/Public/USP/Sistemas%20de%20Informação/2º%20semestre/Matemática%20Discreta%20I/Lista%203/Imagens/79a22d36358fc370cc73cc65d857f9927d7e6fa8.png) | + +### Exercício 7 + +Um dado conjunto A é subconjunto de um conjunto B se A está **contido** em B, isto é, todos os elementos de A também são elementos de B. Os elementos de A e B podendo mesmo coincidir. + +Segue desta definição de subconjunto que o conjunto vazio é contido por todos os conjuntos, pois todos os conjuntos existentes contém os elementos que compõem o conjunto vazio, isto é, nenhum (todos tem nada e mais algo). Ainda, mesmo o conjunto vazio contém todos os elementos que constituem... o conjunto vazio, e portanto também o contém. + +### Exercício 8 + +Uma vez que $A\backslash B = A \cap \overline B$, temos: + +$(A \backslash B) \cup (B \backslash A) += (A \cap \overline B) \cup (B \cap \overline A) = \underbrace{[(A \cap \overline B) \cup B] \cap [(A \cap \overline B) \cup \overline A]}_{\text{Distributiva}} \\ = [(A \cup B) \cap (\overline B \cup B)] \cap [(A \cup \overline A) \cap (\overline B \cup \overline A)] = [(A \cup B) \cap \Omega] \cap [\Omega \cap (\overline B \cup \overline A)]\\ = (A \cup B) \cap (\overline B \cup \overline A) = (A \cup B) \cap \underbrace{(\overline{A \cap B})}_{\text{De Morgan}} = (A \cup B) \backslash (A \cap B)\,\blacksquare$ + +### Exercício 9 + +#### Associatividade na intersecção + +| $B \cap C$ | $A \cap (B \cap C)$ | +|:------------------------------------------------------------------------------------------------------------------------------------------------------------:|:-------------------------------------------------------------------------------------------------------------------------------------------------------------------:| +| ![](/home/user/Public/USP/Sistemas%20de%20Informação/2º%20semestre/Matemática%20Discreta%20I/Lista%203/Imagens/f6718e43f7bfc4d96bf626cbfeaef9a5a00f1257.png) | ![](/home/user/Public/USP/Sistemas%20de%20Informação/2º%20semestre/Matemática%20Discreta%20I/Lista%203/Imagens/5d43aa55b225aa01a83f79681161cddf4d3ee3fe.png) | +| $A \cap B$ | $(A \cap B) \cap C$ | +| ![](/home/user/Public/USP/Sistemas%20de%20Informação/2º%20semestre/Matemática%20Discreta%20I/Lista%203/Imagens/ad71acd12cdaba338848679d7b7f76154e757687.png) | ![](file:///home/user/Public/USP/Sistemas%20de%20Informação/2º%20semestre/Matemática%20Discreta%20I/Lista%203/Imagens/5d43aa55b225aa01a83f79681161cddf4d3ee3fe.png) | + +#### Associatividade na união + +| $B \cup C$ | $A \cup (B \cup C)$ | +|:------------------------------------------------------------------------------------------------------------------------------------------------------------:|:------------------------------------------------------------------------------------------------------------------------------------------------------------:| +| ![](/home/user/Public/USP/Sistemas%20de%20Informação/2º%20semestre/Matemática%20Discreta%20I/Lista%203/Imagens/19640c8af4dfea52049f762f52e639a479e36313.png) | ![](/home/user/Public/USP/Sistemas%20de%20Informação/2º%20semestre/Matemática%20Discreta%20I/Lista%203/Imagens/1d66ddcd5609c366f062eff22f73812fb8282708.png) | +| $A \cup B$ | $(A \cup B) \cup C$ | +| ![](/home/user/Public/USP/Sistemas%20de%20Informação/2º%20semestre/Matemática%20Discreta%20I/Lista%203/Imagens/8f848a759ab6843f845632e171fef607836d22b2.png) | ![](/home/user/Public/USP/Sistemas%20de%20Informação/2º%20semestre/Matemática%20Discreta%20I/Lista%203/Imagens/1d66ddcd5609c366f062eff22f73812fb8282708.png) | + +### Exercício 10 + +#### Distributividade da intersecção na união + +| $B \cup C$ | $A \cap (B \cup C)$ | +|:-------------------------------------------------------------------------------------------------------------------------------------------------------------------:|:------------------------------------------------------------------------------------------------------------------------------------------------------------:| +| ![](file:///home/user/Public/USP/Sistemas%20de%20Informação/2º%20semestre/Matemática%20Discreta%20I/Lista%203/Imagens/19640c8af4dfea52049f762f52e639a479e36313.png) | ![](/home/user/Public/USP/Sistemas%20de%20Informação/2º%20semestre/Matemática%20Discreta%20I/Lista%203/Imagens/99652ebfc205887cb033d4e3d1cbcaee520c9650.png) | + +| $A \cap B$ | $A \cap C$ | $(A \cap B) \cup (A \cap C)$ | +|:------------------------------------------------------------------------------------------------------------------------------------------------------------:|:------------------------------------------------------------------------------------------------------------------------------------------------------------:|:-------------------------------------------------------------------------------------------------------------------------------------------------------------------:| +| ![](/home/user/Public/USP/Sistemas%20de%20Informação/2º%20semestre/Matemática%20Discreta%20I/Lista%203/Imagens/ad71acd12cdaba338848679d7b7f76154e757687.png) | ![](/home/user/Public/USP/Sistemas%20de%20Informação/2º%20semestre/Matemática%20Discreta%20I/Lista%203/Imagens/8dc7516caa67a0946b53c8951a9a9c8ac2d47268.png) | ![](file:///home/user/Public/USP/Sistemas%20de%20Informação/2º%20semestre/Matemática%20Discreta%20I/Lista%203/Imagens/99652ebfc205887cb033d4e3d1cbcaee520c9650.png) | + +| $A \cup B$ | $A \cup C$ | $(A \cup B) \cap (A \cup C)$ | +|:-------------------------------------------------------------------------------------------------------------------------------------------------------------------:|:------------------------------------------------------------------------------------------------------------------------------------------------------------:|:------------------------------------------------------------------------------------------------------------------------------------------------------------:| +| ![](file:///home/user/Public/USP/Sistemas%20de%20Informação/2º%20semestre/Matemática%20Discreta%20I/Lista%203/Imagens/8f848a759ab6843f845632e171fef607836d22b2.png) | ![](/home/user/Public/USP/Sistemas%20de%20Informação/2º%20semestre/Matemática%20Discreta%20I/Lista%203/Imagens/0eb7d5b2aaea2eeec7c705ecabcdd4b00db40016.png) | ![](/home/user/Public/USP/Sistemas%20de%20Informação/2º%20semestre/Matemática%20Discreta%20I/Lista%203/Imagens/38c95bfe63bf5c57d9a87ee53ca75547674b9e99.png) | + +### Exercício 11 + +#### Complemento da intersecção + +| $A \cap B$ | $\overline{A \cap B}$ | +|:----------------------------------------------------------------------------------------------------------------------------------------------------:|:------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------:| +| ![](file:///home/user/Public/USP/Sistemas%20de%20Informação/2º%20semestre/Matemática%20Discreta%20I/Lista%203/Imagens/2021-10-27-23-10-23-image.png) | | + +| $\overline A$ | $\overline B$ | $\overline A \cup \overline B$ | +|:---------------------------------------------------------------------------------------------------------------------------------------------:|:------------------------------------------------------------------------------------------------------------------------------------------------------------:|:------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------:| +| ![](/home/user/Public/USP/Sistemas%20de%20Informação/2º%20semestre/Matemática%20Discreta%20I/Lista%203/Imagens/2021-10-28-13-05-14-image.png) | ![](/home/user/Public/USP/Sistemas%20de%20Informação/2º%20semestre/Matemática%20Discreta%20I/Lista%203/Imagens/57d9f887a8685e25de0394759eccc5808736a5bf.png) | | + +#### Complemento da união + +| $A \cup B$ | $\overline{A \cup B}$ | +|:----------------------------------------------------------------------------------------------------------------------------------------------------:|:------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------:| +| ![](file:///home/user/Public/USP/Sistemas%20de%20Informação/2º%20semestre/Matemática%20Discreta%20I/Lista%203/Imagens/2021-10-27-23-10-06-image.png) | | + +| $\overline A$ | $\overline B$ | $\overline A \cap \overline B$ | +|:----------------------------------------------------------------------------------------------------------------------------------------------------:|:-------------------------------------------------------------------------------------------------------------------------------------------------------------------:|:------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------:| +| ![](file:///home/user/Public/USP/Sistemas%20de%20Informação/2º%20semestre/Matemática%20Discreta%20I/Lista%203/Imagens/2021-10-28-13-05-14-image.png) | ![](file:///home/user/Public/USP/Sistemas%20de%20Informação/2º%20semestre/Matemática%20Discreta%20I/Lista%203/Imagens/57d9f887a8685e25de0394759eccc5808736a5bf.png) | | + +### Exercício 12 + +**a.** $A\Delta B = \{0,1,2,3,7,8,9\}$; + +**b.** $B \Delta C = \{1, 3, 4, 6, 8\}$; + +**c.** $B \Delta D = \{2,3,4,6,9\}; A \cap (B \Delta D) = \{2,3,4,6\}$ + +**d.** $A \cap B = \{4,5,6\}; A \cap D = \{2,3,5\}; (A \cap B) \Delta (A\cap D) = \{2,3,4,6\}$ + +### Exercício 13 + +| $A \Delta B$ | $C$ | $A \Delta B \Delta C$ | +|:------------------------------------------------------------------------------------------------------------------------------------------------------------:|:------------------------------------------------------------------------------------------------------------------------------------------------------------:|:------------------------------------------------------------------------------------------------------------------------------------------------------------:| +| ![](/home/user/Public/USP/Sistemas%20de%20Informação/2º%20semestre/Matemática%20Discreta%20I/Lista%203/Imagens/87ff6311a217e1d31b0034614160ba72bc5bbe40.png) | ![](/home/user/Public/USP/Sistemas%20de%20Informação/2º%20semestre/Matemática%20Discreta%20I/Lista%203/Imagens/1d5ad7dfaf727a225c5b849048ad08571bdbd011.png) | ![](/home/user/Public/USP/Sistemas%20de%20Informação/2º%20semestre/Matemática%20Discreta%20I/Lista%203/Imagens/18801add5f21f8833e5f651b295331072fde9731.png) | + +### Exercício 14 + +**a.** $A \oplus B = (A\backslash B) \cup (B \backslash A) = (B \backslash A) \cup (A\backslash B) = B \oplus A$ + +**b.** $A \oplus B = (A\backslash B) \cup (B \backslash A) = (\overline B\backslash \overline A) \cup (\overline A \backslash \overline B) = (\overline A \backslash \overline B) \cup (\overline B\backslash \overline A) = \overline A \oplus \overline B$ + +**c.** $A \oplus \varnothing = (A \backslash \varnothing) \cup (\varnothing \backslash A) = A \cup \varnothing = A$ + +**d.** $A \oplus A = (A\backslash A) \cup (A\backslash A) = \varnothing \cup \varnothing = \varnothing$ + +**e.** $A * A = A \cap A = A$ + +**f.** $A \oplus (B \oplus C) = A \oplus B \oplus C = (A \oplus B) \oplus C$ + +**g.** $A \oplus B = A \oplus C \implies (A \oplus B) \cap \overline A = (A \oplus C) \cap \overline A \implies (B \backslash A) = (C \backslash A)$ + +Pela definição de diferença, tem-se que $B\backslash A = \{x : (x \in B) \land (x \not \in A) \}$, e $C\backslash A = {x : (x \in C) \land (x \not \in A) }$. Ora, se $B\backslash A$ equivale a dizer que um elemento está em $C$ mas não em $A$ ($C\backslash A$), então $B = C$. + +**h.** $A * (B \oplus C) = A \cap (B \oplus C) = \underbrace{(A \cap B) \oplus (A \cap C)}_{\text{Distributividade na intersecção}} = (A * B) \oplus (A * C)$ + +Propriedade esta da distributividade demonstrada no exercício 10. + +### Exercício 15 + +**a.** Cada subconjunto a integrar o produto fundamental pode assumir 2 formas distintas: $A_i$ ou $\overline A_i$. Assim sendo, conforme a análise combinatória, para $n$ subconjuntos existem $2_1 \times 2_2 \times \dots \times 2_n = 2^n$ possibilidades distintas de produto fundamental. + +**b.** Segue da formulação anterior que, para cada par $J$ e $K$ de produto fundamental existe pelo menos um conjunto $\overline A_i$ ($1 \le i \le n$) em $J$ que é complementar ao conjunto $A_i$ em $K$. Isto é, dado um elemento $x$ qualquer tem-se: + +$$ +\{x : (x \in A_i)\ \underline \lor\ (x \in \overline A_i)\} +$$ + +Onde $\underline \lor$ é o "ou exclusivo". Como $J = \{x : x \in (A_1 \cap \dots \cap A_i \cap \dots \cap A_n)\}$, $K = \{x : x \in (A_1 \cap \dots \cap \overline A_i \cap \dots \cap A_n)\}$ e a definição de intersecção para quaisquer conjuntos $A$ e $B$ é $A \cap B = \{x : (x \in A) \land (x \in B)\}$, não à elemento em $J$ que também pertença à $K$. Estes conjuntos são, portanto, **disjuntos** entre si. + +**c.** O conjunto Universo $\Omega$ é aquele que engloba a todos os elementos que pertencem à qualquer conjunto. Consideremos o par de conjuntos $J$ e $K$ anterior. Um elemento $x$ que pertence a $J$ não pertence a $K$ e vice-versa, não obstante este pertence a algum conjunto e portanto pertence também ao conjunto Universo. Pela definição de produto fundamental, podemos extrapolar essa relação para qualquer número $n$ de conjuntos de produto fundamental. Assim, qualquer elemento $x$ é tal que pertence a um produto fundamental, não pertence aos $(n - 1)$ demais, e pertence ao conjunto Universo. + +Como todos os elementos $x$ são assim compreendidos pelo conjunto Universo, pela definição de subconjunto dada no exercício 6, todo produto fundamental é subconjunto do conjunto Universo e, por conseguinte, o conjunto Universo unifica todos os produtos fundamentais. + +## Exercício 16 + +Um subconjunto $X$ de $S$ é tal que possui $i$ elementos, $0 \le i \le n$, deste último. Ou seja, para cada elemento de $S$ existem 2 possibilidades: estar ou não em $X$. Assim sendo, conforme a análise combinatória, para $n$ elementos existem $2_1 \times 2_2 \times \dots \times 2_n = 2^n$ possíveis subconjuntos. + +Tal qual fizemos no exercício 4, podemos quantificar o número de subconjuntos a conter $m$ elementos pela seguinte relação binominal: + +$$ +\binom nm = \frac{n!}{m!(n - m)!} +$$ + +### Exercício 17 + + + +A relação dada pelo enunciado trata-se do *Princípio de Inclusão e Exclusão*. + +$$ +|A \cup B| = |A| + |B| - |A \cap B| +$$ + +De fato, ao contarmos o número de elementos em $|A \cup B|$ pela soma dos elementos em $|A| + |B|$, necessitamos também subtrair o número de elementos em $|A \cap B|$ de forma a evitar que estes sejam contabilizados duas vezes. + +### Exercício 18 + +Uma forma mais geral do Princípio de Inclusão Exclusão pode ser expressa como: + +$$ +\left | \bigcup^n_{i = 1} A_i \right | = \sum^n_{i = 1} |A_i| +- \sum^{n - 1}_{i = 1}\sum^n_{j = i + 1} | A_i \cap A_j | ++ (-1)^{n - 1} \left | \bigcap^n_{i = 1} A_i \right | +$$ + +Assim, temos que + +$|A \cup B \cup C| = (|A| + |B| + |C|) - (|A \cap B| + |A \cap C| + |B \cap C|) + |A \cap B \cap C|$ + +Enquanto + +$|A \cup B \cup C \cup D| = (|A| + |B| + |C| + |D|) - (|A \cap B| + |A \cap C| + |A \cap D| \\ + |B \cap C| + |B \cap D| + |C \cap D|) - |A \cap B \cap C \cap D|$ + +### Exercício 19 + +Para qualquer elemento $x$, se $x$ está contido em $A$ este + +- está contido em um subconjunto de $P(A)$ + +- não está contido em $B$ ou um subconjunto de $P(B)$ + +e vice-versa. Isso pois $A$ e $B$ tratam-se de conjuntos **disjuntos**. Assim sendo, + +- $P(A) \cap P(B) = A \cap B = \varnothing$ + +- $P(A) \cup P(B) \subseteq A \cup B$ + +### Exercício 20 + +Observe o seguinte gráfico: + +```vega-lite +{ + "$schema": "https://vega.github.io/schema/vega/v5.json", + "description": "A basic stacked bar chart example.", + "width": 500, + "height": 200, + "padding": 5, + + "data": [ + { + "name": "table", + "values": [ + {"x": "São religiosos", "y": 79, "c": 0}, {"x": "São religiosos", "y": 21, "c": 1}, + {"x": "Fizeram compras online", "y": 26, "c": 0}, {"x": "Fizeram compras online", "y": 74, "c": 1} + ], + "transform": [ + { + "type": "stack", + "groupby": ["x"], + "sort": {"field": "c"}, + "field": "y" + } + ] + } + ], + + "scales": [ + { + "name": "x", + "type": "band", + "range": "width", + "domain": {"data": "table", "field": "x"} + }, + { + "name": "y", + "type": "linear", + "range": "height", + "nice": true, "zero": true, + "domain": {"data": "table", "field": "y1"} + }, + { + "name": "color", + "type": "ordinal", + "range": "category", + "domain": {"data": "table", "field": "c"} + } + ], + + "axes": [ + {"orient": "bottom", "scale": "x", "zindex": 1}, + {"orient": "left", "scale": "y", "zindex": 1} + ], + + "marks": [ + { + "type": "rect", + "from": {"data": "table"}, + "encode": { + "enter": { + "x": {"scale": "x", "field": "x"}, + "width": {"scale": "x", "band": 1, "offset": -1}, + "y": {"scale": "y", "field": "y0"}, + "y2": {"scale": "y", "field": "y1"}, + "fill": {"scale": "color", "field": "c"} + }, + "update": { + "fillOpacity": {"value": 1} + }, + "hover": { + "fillOpacity": {"value": 0.5} + } + } + } + ] +} +``` + +Se admitirmos que o maior número possível de pessoas não religiosas e que nunca fizeram uma compra online, temos que no mínimo o número de pessoas religiosas que nunca fizeram compras é $[100 - (26 + 21)]\% = 53\%$ da população. Por outro lado, + +```vega-lite +{ + "$schema": "https://vega.github.io/schema/vega/v5.json", + "description": "A basic stacked bar chart example.", + "width": 500, + "height": 200, + "padding": 5, + + "data": [ + { + "name": "table", + "values": [ + {"x": "São religiosos", "y": 79, "c": 0}, {"x": "São religiosos", "y": 21, "c": 1}, + {"x": "Não fizeram compras online", "y": 74, "c": 0}, {"x": "Não fizeram compras online", "y":26, "c": 1} + ], + "transform": [ + { + "type": "stack", + "groupby": ["x"], + "sort": {"field": "c"}, + "field": "y" + } + ] + } + ], + + "scales": [ + { + "name": "x", + "type": "band", + "range": "width", + "domain": {"data": "table", "field": "x"} + }, + { + "name": "y", + "type": "linear", + "range": "height", + "nice": true, "zero": true, + "domain": {"data": "table", "field": "y1"} + }, + { + "name": "color", + "type": "ordinal", + "range": "category", + "domain": {"data": "table", "field": "c"} + } + ], + + "axes": [ + {"orient": "bottom", "scale": "x", "zindex": 1}, + {"orient": "left", "scale": "y", "zindex": 1} + ], + + "marks": [ + { + "type": "rect", + "from": {"data": "table"}, + "encode": { + "enter": { + "x": {"scale": "x", "field": "x"}, + "width": {"scale": "x", "band": 1, "offset": -1}, + "y": {"scale": "y", "field": "y0"}, + "y2": {"scale": "y", "field": "y1"}, + "fill": {"scale": "color", "field": "c"} + }, + "update": { + "fillOpacity": {"value": 1} + }, + "hover": { + "fillOpacity": {"value": 0.5} + } + } + } + ] +} +``` + +Se admitirmos que a correspondência entre pessoas religiosas e que não fizeram compras online é máxima, teremos que todos que não fizeram compras online, $74\%$ da população, são religiosos. + +Por isso esse índice nunca é igual ou inferior à 50% da população. + +### Exercício 21 + +Todos aqueles múltiplos de $2 \times 3 = 6$ e $2 \times 5 = 10$, descontados aqueles múltiplos de $2 \times 3 \times 5 = 30$. 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c;UB)3ZDzb)dq|f3`uiV^fG1j1;5<(M4P8ZY<^TWy literal 0 HcmV?d00001 diff --git a/Matemática Discreta I/Lista 4/Resolução Lista 4.1 - Relações - MD.md b/Matemática Discreta I/Lista 4/Resolução Lista 4.1 - Relações - MD.md new file mode 100644 index 0000000..fb7eb49 --- /dev/null +++ b/Matemática Discreta I/Lista 4/Resolução Lista 4.1 - Relações - MD.md @@ -0,0 +1,275 @@ +# Resolução da [Lista 4](https://drive.google.com/file/d/1ls29hxpLdYCc-sF8LkKYiFCs1M79PUKb/view?usp=drive_web&authuser=0) da disciplina de Matemática Discreta + +> Feita por Guilherme de Abreu Barreto[^1] + +## Relações + +### Exercício 1 + +Se concordarmos que a Teoria dos Conjuntos provêm uma sólida fundamentação axiomática a partir da qual construirmos demais saberes matemáticos, então temos de demonstrar como demais objetos matemáticos podem ser descritos enquanto conjuntos de algum tipo. Ou seja, se **pares ordenados** não forem compreendidos enquanto axiomas, então estes podem ser descritos enquanto conjuntos. O principal problema o qual temos de sanar nesta representação é o fato de que conjuntos descrevem qualquer agrupamento de elementos distintos, mesmo aqueles **desordenados**; tal que um conjunto$A = \{a,b\} = \{b,a\}$. + +Para sanar essa insuficiência, o matemático Kazimierz Kuratowski propôs em 1921 a seguinte definição: + +- Considere um conjunto com dois valores $a, b$: + +$A = \{a,b\}$ + +- Então, o conjunto potência de $A$ é: + +$P(A) = \{\{\}, \{a\},\{b\},\{a,b\}\}$ + +- Se deste conjunto derivarmos um subconjunto contendo todos os elementos que por vez contêm $a$, teremos: + +$S(P(A)) = \{\{a\},\{a,b\}\}$ + +Note que este subconjunto contém toda informação necessária para descrevermos um par ordenado: + +- Os valores $a$ e $b$. + +- A ordenação destes: o primeiro elemento é descrito pelo elemento $\{a\}$ + +Finalmente, podemos então restituir a notação original estabelecendo a correspondência $(a,b) := \{\{a\},\{a,b\}\}$. + +Voltemos ao problema em questão. $(a,b)$ = $(c,d)$ se e somente se $a = c$ e $b = d$. Procederemos na ida por demonstração direta: + +$$ +(a,b) := \{\{a\},\{a,b\}\} = \{\{c\},\{c,d\}\} := (c,d) +$$ + +E na volta procederemos por contradição. Por hipótese, a = c e b = d, onde $(a,b) := \{\{a\},\{a,b\}\}$. Vamos assumir aqui que também a = b = c = d. Pela definição de conjuntos, os elementos que constituem um conjunto são todos **distintos entre si**, de tal forma que repetições são redundantes, não constituem novos elementos. Assim: + +$$ +(a,a) := \{\{a\}, \{\underbrace{a,a}_{ =\ a}\}\} += \{\underbrace{\{a\},\{a\}}_{=\ \{a\}}\} = \{\{a\}\} +$$ + +Podemos notar que a cardinalidade deste conjunto é distinta daquela da hipótese: + +$|\{\{a\}\}| = 1;\\ |\{\{a\},\{a,b\}\}| = 2$ + +O que é absurdo. Logo, só é possível que $(a,b) = (c,d)$ se $a = b$ e $c = d$, e não doutra forma. $\blacksquare$ + +### Exercício 2 + +**a.** + + + +**b.** + +| R | a | b | c | +| ----- | --- | --- | --- | +| **1** | 0 | 1 | 1 | +| **2** | 0 | 0 | 0 | +| **3** | 0 | 1 | 0 | +| **4** | 1 | 0 | 1 | + +**c.** $R^{-1} = \{(b,1),(c,1),(b,3),(a,4),(c,4)\}$ + +| R^-1^ | 1 | 2 | 3 | 4 | +| ----- | --- | --- | --- | --- | +| **a** | 0 | 0 | 0 | 1 | +| **b** | 1 | 0 | 1 | 0 | +| **c** | 1 | 0 | 0 | 1 | + +**d.** Uma é a matriz transposta da outra. Isso é uma se obtêm pela transposição dos elementos ordenados em linhas na outra para colunas na própria. + +### Exercício 3 + + + +**a.** $T = R \circ S = \{(1,x), (2,y), (2,z)\}$ + +**b.** $M_R = \left[\begin{matrix} +0 & 1 & 0 \\ +1 & 0 & 1 \\ +0 & 0 & 0 \\ +0 & 0 & 0 \end{matrix}\right]; M_S = \left[\begin{matrix} +0 & 1 & 0 \\ +1 & 0 & 0 \\ +0 & 1 & 1 \end{matrix}\right]$ + +**c.** Estas são ligeiramente diferentes: + +$M_T = \left[\begin{matrix} +1 & 0 & 0 \\ +0 & 1 & 1 \\ +0 & 0 & 0 \\ +0 & 0 & 0 \end{matrix}\right] \\\ \\ +M_RM_S = \left[\begin{matrix} +0 & 1 & 0 \\ +1 & 0 & 1 \\ +0 & 0 & 0 \\ +0 & 0 & 0 \end{matrix}\right] \cdot \left[\begin{matrix} +0 & 1 & 0 \\ +1 & 0 & 0 \\ +0 & 1 & 1 \end{matrix}\right] = +\left[\begin{matrix} +1 & 0 & 0 \\ +0 & 2 & 1 \\ +0 & 0 & 0 \\ +0 & 0 & 0 \end{matrix}\right]$ + +Note que a matriz resultante $M_RM_S$ corresponde às posições de incidência das setas em $C$ e também o número destas: + +| T = R ○ S | x | y | z | +| --------- | --- | --- | --- | +| **1** | 1 | 0 | 0 | +| **2** | 0 | 2 | 1 | +| **3** | 0 | 0 | 0 | +| **4** | 0 | 0 | 0 | + +### Exercício 4 + +| R | S | R ○ S | +|:---------------------------------------------------------------------------------------------------------------------------------------------:|:----------------------------------------------------------------------------------------------------------------------------------------------------:|:---------------------------------------------------------------------------------------------------------------------------------------------:| +| ![](/home/user/Public/USP/Sistemas%20de%20Informação/2º%20semestre/Matemática%20Discreta%20I/Lista%204/Imagens/2021-10-30-18-31-10-image.png) | ![](/home/user/Public/USP/Sistemas%20de%20Informação/2º%20semestre/Matemática%20Discreta%20I/Lista%204/Imagens/2021-10-30-18-51-16-image.png) | ![](/home/user/Public/USP/Sistemas%20de%20Informação/2º%20semestre/Matemática%20Discreta%20I/Lista%204/Imagens/2021-11-03-15-46-10-image.png) | +| **S ○ R** | **R^2^** | **S^2^** | +| ![](/home/user/Public/USP/Sistemas%20de%20Informação/2º%20semestre/Matemática%20Discreta%20I/Lista%204/Imagens/2021-11-03-15-20-33-image.png) | ![](file:///home/user/Public/USP/Sistemas%20de%20Informação/2º%20semestre/Matemática%20Discreta%20I/Lista%204/Imagens/2021-10-30-18-31-10-image.png) | ![](/home/user/Public/USP/Sistemas%20de%20Informação/2º%20semestre/Matemática%20Discreta%20I/Lista%204/Imagens/2021-10-30-18-59-26-image.png) | + +### Exercício 5 + +| R^-1^ | R^2^ | +|:---------------------------------------------------------------------------------------------------------------------------------------------:|:---------------------------------------------------------------------------------------------------------------------------------------------:| +| ![](/home/user/Public/USP/Sistemas%20de%20Informação/2º%20semestre/Matemática%20Discreta%20I/Lista%204/Imagens/2021-10-30-19-23-15-image.png) | ![](/home/user/Public/USP/Sistemas%20de%20Informação/2º%20semestre/Matemática%20Discreta%20I/Lista%204/Imagens/2021-10-30-19-27-23-image.png) | + +Nota-se que as relações $R$, $R^{-1}$ e $R^2$ **não** são funções pois mapeiam pelo menos um valor no domínio para mais de um valor na imagem. + +### Exercício 6 + +| R | R^-1^ | +|:---------------------------------------------------------------------------------------------------------------------------------------------:|:---------------------------------------------------------------------------------------------------------------------------------------------:| +| ![](/home/user/Public/USP/Sistemas%20de%20Informação/2º%20semestre/Matemática%20Discreta%20I/Lista%204/Imagens/2021-10-30-19-52-13-image.png) | ![](/home/user/Public/USP/Sistemas%20de%20Informação/2º%20semestre/Matemática%20Discreta%20I/Lista%204/Imagens/2021-10-30-19-54-07-image.png) | +| **R ∪ R^-1^** | **R ∩ R^-1^** | +| ![](/home/user/Public/USP/Sistemas%20de%20Informação/2º%20semestre/Matemática%20Discreta%20I/Lista%204/Imagens/2021-10-30-19-54-52-image.png) | ![](/home/user/Public/USP/Sistemas%20de%20Informação/2º%20semestre/Matemática%20Discreta%20I/Lista%204/Imagens/2021-10-30-19-56-21-image.png) | + +### Exercício 7 + +Sejam R e S ordens parciais sobre um conjunto A. Mostre que R∩S também é uma relação de ordem parcial sobre A. + +Para uma relação ser de tipo ordem parcial, esta necessita ser **reflexiva**, **antissimétrica** e **transitiva**. A relação de intersecção $R \cap S$ é deste tipo por consequência de carregar tais características dos conjuntos que integra: + +- Esta é reflexiva pois qualquer par ordenado $(a,a) \in R$ corresponde s $(a,a) \in S$ e portanto a $(a,a) \in R \cap S$. + +- Para qualquer par ordenado $(a,b) \in R \cap S$, $(a,b) \in R$ e $(a,b) \in S$. Como tanto $R$ e $S$ são antissimétricos, $(b,a) \not\in R$, $(b,a) \not\in S$ e $(b,a) \not\in R \cap S$. + +- Para qualquer par ordenado $(a,b), (b,c) \in R \cap S$, $(a,b), (b,c) \in R$ e $(a,b), (b,c) \in S$. Como tanto R e S são transitivos, $(a,c) \in R$, $(a,c) \in S$ e $(a,c) \in R \cap S$. + +### Exercício 8 + +Tal relação é: + +- Reflexiva pois $a^1 = a$, $1 \in \N$. Portanto $(a,a) \in R$. + +- Antissimétrica pois se $a^r = b$ então $b^\frac 1r = \pm a$, mas $\frac 1r \not \in \N$. Portanto $(a,b) \in R$, mas $(b, a) \not \in R$. + +- Transitiva pois se $a^r = b$ e $b^s = c$ então $a^{r \cdot s} = c$, $r \cdot s \in \N$. Então $(a,c) \in R$. + +O conjunto destas qualidades configura que o conjunto $R$ é de tipo parcialmente ordenado sobre $\Z$. + +### Exercício 9 + +Tal relação é: + +- Reflexiva pois se $A = B$ então $(A,A) \in R$. + +- Antissimétrica pois se $A \subset B$ então $B \not \subset A$. Portanto $(A,B) \in R$, mas $(B, A) \not \in R$. + +- Transitiva pois se $A \subseteq C$ e $C \subseteq B$ então $A \subseteq B$. Logo $(A,C) \in R$, $(C, B) \in R$ e $(A, B) \in R$. + +O conjunto destas qualidades configura que o conjunto R é de tipo parcialmente ordenado. + +**Diagrama de Hasse para S = P({a,b,c})** + + + +### Exercício 10 + +Uma relação de equivalência trata-se de uma relação binária que é **reflexiva**, **simétrica** e **transitiva**. + +Assim sendo, a relação descrita é de tal tipo tido que ela é + +- Reflexiva pois se $p = r$ e $q = s$ então $((p,q)(p,q)) \in R$. + +- Simétrica pois se $pq = rs$ então $((p,q)(r,s)) \in R$ e $((r,s),(p,q)) \in R$. + +- Transitiva pois se $pq = mn$ e $mn = rs$ então $((p,q),(m,n)) \in R$, $((m,n),(r,s)) \in R$ e $((p,q),(r,s)) \in R$. + +### Exercício 11 + +**(Ida)** A cardinalidade do conjunto $|\Z|$ é infinita, entretanto $P(\Z)$ contém todos os subconjuntos possíveis de serem compostos por elementos de $\Z$, tal que $P(\Z) = \{\{\},\{0\},\{1\},\{-1\},\{2\},\dots,\{0,1\},\{0, -1\},\dots\}$. Assim $P(\Z)$ contém incontáveis subconjuntos cada qual finito, tal que para dois subconjuntos $A$ e $B$ quaisquer a diferença simétrica entre estes também é finita pois $A \Delta B \subseteq A\cup B$. + +**(Volta)** A relação $R$ entre quaisquer conjuntos finitos é + +- Reflexiva pois $A \Delta A = \varnothing$, um conjunto finito contendo nenhum elemento. Logo, $(A,A) \in R$. + +- Simétrica pois $A \Delta B = B \Delta A$. Assim, $(A, B) \in R$ e $(B, A) \in R$. + +- Transitiva pois se $A \Delta B$ é finita e $B \Delta C$ também, isso implica que $A \Delta C$ também será. Portanto $((A, B), (B,C) \in R) \implies ((A,C) \in R)$. + +Finalmente, observa-se que $R$ trata-se de uma relação de **equivalência**. + +### Exercício 12 + +Temos a relação $R = \{(a,b) \in \R : (b - a) \in \Z\}$, está é uma relação de equivalência pois + +- esta é reflexiva: para qualquer $a \in \R$, $(a - a) = 0$ e $0 \in \Z$, logo $(a,a) \in R$; + +- esta é simétrica: pois o módulo da diferença de $b - a$ equivale àquele da diferença de $a - b$ em $\Z$, logo $(a,b), (b,a) \in R$; + +- esta é transitiva pois se $(a - b), (b - c) \in \Z$ então $(a - c) \in \Z$, logo $(a,b),(b,c),(a,c) \in R$. + +Assim, para qualquer valor $x \in \R$ dada a relação $R$ sobre $\R$ tem-se a classe de equivalência $[x] = \{y \in \R : (x.y) \in R \}$ representativa de todos os valores aqueles para os quais a diferença $x - y$ produz um número inteiro. + +### Exercício 13 + +- $R$ e $R^{-1}$ são relações transitivas tais que $(a,b),(b,c),(a,c) \in R$ e $(c,b),(b,a),(c,a) \in R^{-1}$, logo $R \cap R^{-1}$ também é transitiva pois $(a,b),(b,c),(a,c), (c,b),(b,a),(c,a) \in R \cap R^{-1}$; + +- $R$ e $R^{-1}$ são relações reflexivas, então $(a,a) \in R$, $(a,a) \in R^{-1}$ e portanto $(a,a) \in R \cap R^{-1}$; + +- Finalmente, se $(a,b)$ e $(b,a)$ está em $R \cap R^{-1}$, tal qual demonstrado anteriormente, então $R \cap R^{-1}$ é também simétrica e constitui uma relação de equivalência. + +### Exercício 14 + +A relação $a \equiv b(\text{mod }n)$ denota existência da igualdade $a = kn + b$ para algum $k \in \Z^*$. Podemos notar que esta trata-se de uma relação de equivalência pois esta possui as características de + +- reflexividade: existe $a \equiv a(\text{mod }n)$, para qualquer $n$ quando $a = 0$ e para $a$ quando $n > a$; + +- simetria: $a \equiv b(\text{mod }n)$ se, e somente se, $b \equiv a(\text{mod }n)$. + +$$ +a = kn + b \iff b = a\ \underbrace{- k}_{k_2}n = k_2n + a +$$ + +- transitividade: se $a \equiv b(\text{mod }n)$ e $b \equiv c(\text{mod }n)$, então $a \equiv c(\text{mod }n)$. + +$\begin{cases} a = kn + b \\ b = k_2n + c \end{cases} \\ \therefore +a = \underbrace{kn + k_2n}_{k_3n} + c = k_3n + c$ + +### Exercício 15 + +Sejam $A, B, C$ matrizes de dimensão $n \times n$ e $P$ uma matriz inversível também de dimensão $n \times n$ tal que duas matrizes similares entre si, denotadas por $A \sim B$, estão relacionadas por $PAP^{-1} = B$. Esta relação de similitude trata-se de uma relação de equivalência pois: + +- Esta é reflexiva: $A \sim A$ + +**Prova:** Seja $P$ a matriz identidade $I_n$, $(I_n)^{-1}AI_n = A$. $\blacksquare$ + +- Esta é simétrica: $A \sim B$ se e apenas se $B \sim A$ + +**Prova:** se assumirmos que $A \sim B$, teremos + +$P^{-1}AP = B\\ \cancel{P(P^{-1}}AP) = P(B) \\ (A\cancel{P)P^{-1}} = PBP^{-1} \\ PBP^{-1} = A$ + +Seja $Q$ uma matriz tal que $Q = P^{-1}$, logo $Q^{-1}BQ = A \implies B \sim A$ + +Assim, $A \sim B \iff B\sim A\ \blacksquare$. + +- Esta é transitiva: se $A \sim B$ e $B \sim C$ então $A \sim C$ + +**Prova:** Por hipótese temos + +$\begin{cases} B = P^{-1}AP \\ C = Q^{-1}BQ \end{cases} \\ \therefore C = Q^{-1}(P^{-1}AP)Q = (PQ)^{-1}A(PQ)$ + +Seja $W$ uma matriz tal que $W = PQ$, logo $C = W^{-1}A W \implies C \sim A\ \blacksquare$ + +[^1]: nUSP: 12543033; Turma 04 \ No newline at end of file diff --git a/Matemática Discreta I/Lista 4/Resolução Lista 4.2 - Funções - MD.md b/Matemática Discreta I/Lista 4/Resolução Lista 4.2 - Funções - MD.md new file mode 100644 index 0000000..09fd165 --- /dev/null +++ b/Matemática Discreta I/Lista 4/Resolução Lista 4.2 - Funções - MD.md @@ -0,0 +1,217 @@ +# Resolução da [Lista 4](https://drive.google.com/file/d/1ls29hxpLdYCc-sF8LkKYiFCs1M79PUKb/view?usp=drive_web&authuser=0) da disciplina de Matemática Discreta + +> Feita por Guilherme de Abreu Barreto[^1] + +## Funções + +### Exercício 1 + +**a.** Não pois, existem mulheres solteiras as quais, por definição, não são contempladas por esta relação. + +**b.** Está bem definida. + +### Exercício 2 + +Vamos admitir dois conjuntos quaisquer $X$, de tamanho $n$, e $Y$ de tamanho $m$, sendo $m \ge n$, e $n, m \in \N$. Estes estão relacionados entre si pela função injetora $f: X \to Y$. Então, pela definição de função injetora, para quaisquer elementos $x_i, x_j$, $1 \le i < j \le n$, em $X$ para os quais $x_i \not = x_j$ correspondem dois valores $f(x_i)$ e $f(x_j)$ em $Y$ os quais $f(x_i) \not = f(x_j)$. Logo, segue que a mesma função no sentido inverso $f^{-1} : Y \to X$ produz um pareamento de um para um tal que para quaisquer valores $f(x_i) \not = f(x_j)$ resultam valores $x_i \not = x_j$. + +No mais, a função composta $f^{-1} \circ f$ comporta-se tal qual uma função identidade $\text{id}_n$ para $X$ se e somente se $m \ge n$. Por um lado, esta mapeia um valor $x_i$ com ele próprio: + +$$ +f^{-1} \circ f (x_i) = f^{-1}(f(x_i)) = x_i +$$ + +Por outro, isso só é possível para valores de $i \le n$ pois $(x_{n+1},f(x_{n + 1})) \not \in f$. + +### Exercício 3 + +Vamos admitir que os conjuntos $X$, $Y$ e $Z$ possuam tamanhos $r$, $s$ e $t$. Ainda, que os índices $i$, $j$, $k$ são tais que $1 \le k \le t \le j \le s \le i \le r$, sendo $i,j,k,r,s,t \in \N$. Assim, $X = \{x_1, \dots \, x_r\}$, $Y = \{y_1, \dots , y_s\}$, $Z = \{z_1, \dots , z_t\}$. Segue da definição de sobrejeção que + +- $\exists x_i \in X$ tal que $f(x_i) = y_j$ sendo $y_j \in Y$. Ainda, $Y = \{f(x_1), \dots, f(x_r)\}$. + +- $\exists y_j \in Y$ tal que $g(y_j) = z_k$ sendo $z_k \in Z$. Ainda, $Z = \{g(y_1), \dots, g(y_s)\}$. + +Assim, se aplicarmos a função $g \circ f$ sobre $X$ teremos: + +$\{g \circ f(x_1), \dots, g \circ f(x_r)\} = +\{g (f(x_1)), \dots, g (f(x_r))\} = +\{g (y_1), \dots, g (y_s)\} = \\ +\{z_1, \dots, z_t\} = Z$ + +Ou seja $g \circ f$ também é sobrejetora ao mapear $g \circ f: X \to Z$. + +### Exercício 4 + +Quando dizemos que $(x + 1)^2 = x^2 + 2x + 1$ estamos indicando que **para qualquer** número $x$ tais expressões possuem o mesmo valor. Isso também pode ser interpretado dizendo que ambos os lados da igualdade representam **a mesma função**. + +Retomando o problema em questão, temos que $g \circ f = h \circ f \implies g(f(x)) = h(f(x)), f(x) \in Y$. Como $f$ é sobrejetora não existe elemento em $Y$, domínio tanto de $g$ e $h$, o qual não possa ser descrito na forma $f(x)$. Assim sendo, se $g(f(x)) = h(f(x))$ para qualquer valor $f(x)$, por definição estamos falando de funções iguais entre si. + +### Exercício 5 + +Em concordância com a definição de função um grafo orientado é adequado a representação de função se e somente se este indica relações entre pares ordenados. Por exemplo: + +| Representativo de uma função | Não representativo de uma função | +|:------------------------------------------------------------------------------------------------------------------------------------------------------------:|:------------------------------------------------------------------------------------------------------------------------------------------------------------:| +| ![](/home/user/Public/USP/Sistemas%20de%20Informação/2º%20semestre/Matemática%20Discreta%20I/Lista%204/Imagens/e665c17467939fa319f8cd06741fbab4109a1000.png) | ![](/home/user/Public/USP/Sistemas%20de%20Informação/2º%20semestre/Matemática%20Discreta%20I/Lista%204/Imagens/0d0ace6839a652e2515cebb632b852b8ee4e0334.png) | + +No primeiro gráfico vemos que para qualquer nó $x$ à uma relação com um único $f(x)$. No segundo gráfico, um **grafo acíclico dirigido**, isso não ocorre: podemos destacar a relação $f(11) = \{2, 9, 10\}$. + +### Exercício 6 + +**a.** A função $f: \Z \to \Z$ pode ser definida como + +$$ +f = \{(x,y): x = 2k \to y = 1\ \underline \lor\ x = 2k + 1 \to y = -1, \forall k \in \Z\} +$$ + +Onde $\underline \lor$ é o sinal para a expressão "ou exclusivo" + +**b.** Procederemos por exaustão. + +- $z_1$ par e $z_2$ par implicam $z_1 + z_2$ par. + + $$ + 2k + 2k_2 = 2(k + k_2) = 2k_3 + $$ + + Logo, + + $$ + f(z_1 + z_2) = 1 = 1 \cdot 1 = f(z_1)f(z_2) + $$ + +- $z_1$ par e $z_2$ ímpar, ou vice versa, implica $z_1 + z_2$ ímpar. + + $$ + 2k + 2k_2 + 1 = 2(k + k_2) + 1 = 2k_3 + 1 + $$ + + Logo, + + $$ + f(z_1 + z_2) = -1 = -1 \cdot 1 = f(z_1)f(z_2) + $$ + +- $z_1$ e $z_2$ impares implicam, $z_1 + z_2$ par. + + $$ + 2k + 1 + 2k_2 + 1 = 2(k + k_2 + 1) = 2k_3 + $$ + + Logo, + + $$ + f(z_1 + z_2) = 1 = -1 \cdot -1 = f(z_1)f(z_2) + $$ + +**c.** Procederemos por exaustão. + +- $z_1$ par e $z_2$ par implicam $z_1z_2$ par. + + $$ + 2k \cdot 2k_2 = 2(2kk_2) = 2k_3 + $$ + + Logo, + + $$ + f(z_1z_2) = 1 = 1 \cdot 1 = f(z_1)f(z_2) + $$ + +- $z_1$ par e $z_2$ ímpar, ou vice versa, implica $z_1z_2$ par. + + $$ + 2k(2k_2 + 1) = 2[k(2k_2 + 1)] = 2k_3 + $$ + + Logo, + + $$ + f(z_1z_2) = 1 \not = -1 \cdot 1 = f(z_1)f(z_2) + $$ + +- $z_1$ e $z_2$ impares implicam, $z_1z_2$ ímpar. + + $$ + (2k + 1)(2k_2 + 1) = 4kk_2 + 2k + 2k_2 + 1 = 2(2kk_2 + k + k_2) + 1 = 2k_3 + 1 + $$ + + Logo, + + $$ + f(z_1z_2) = - 1 \not = -1 \cdot -1 = f(z_1)f(z_2) + $$ + + **d.** $f(x) = 1$. + + ### Exercício 7 + + **a.** $f_{c,d} \circ f_{a,b}(x) = f_{c,d}(f_{a,b}(x)) = c(ax + b) + d = \underbrace{(ca)}_{=\ p}x + \underbrace{(cb + d)}_{=\ q} = f_{p,q} (x)$ + + **b.** $\big \{ f_{a,b} \circ f_{c,d}(x) = a(cx + d) + b = (ca)x + (ad + b) \\ \therefore f_{a,b} \circ f_{c,d}(x) = f_{c,d} \circ f_{a,b}(x) \implies \cancel{(ca)x} + (cb + d) = \cancel{(ca)x} + (ad + b)\\ \implies b(c - 1) = d(a - 1)$ + + **c.** $f_{a,b} \circ f_{1,1} = f_{1,1} \circ f_{a,b} \implies a(x + 1) + b = 1(ax + b) + 1\\ \implies \cancel{ax} + \cancel b + 1 = \cancel{ax} + a + \cancel b \implies a = 1$ + + Assim sendo, desde que $a = 1$ esta expressão é verdadeira $\forall b \in \R$. + + **d.** Sendo $y = f_{a,b}(x) = ax + b$, a função inversa pode ser expressa por: + + $$ + x = ay + b \implies f^{-1}_{a,b} (x) = \frac{b - x}a + $$ + + ### Exercício 8 + + Para a função + + ```c + long long unsigned int ackermann (unsigned int m, unsigned int n) { + if (m == 0) + return n + 1; + if (n == 0) + return ackermann (m - 1, 1); + return ackermann (m - 1, ackermann(m, n - 1)); + } + ``` + + Os resultados foram, respectivamente: + + ``` + A(1,1) = 3 + A(1,2) = 4 + A(2,2) = 7 + A(3,2) = 29 + ``` + + Para o valor $A(4,2)$ o algoritmo foi executado até que a memória a este alocada fosse esgotada (`segmentation fault`) . Não obstante, conforme constata o artigo referente ao algoritmo na Wikipédia, computadores otimizados para esta tarefa calcularam o resultado de 19,729 dígitos decimais: 2^65536^ − 3. + + ### Exercício 9 (*Divertissement*) + + Consideremos uma lista exaustiva dos infinitos números entre 0.0 e 1.0: + + | 0. | 1 | 1 | 1 | 1 | 1 | 1 | 1 | $\dots$ | + | -------- | -------- | -------- | -------- | -------- | -------- | -------- | -------- | -------- | + | 0. | 1 | 0 | 0 | 0 | 0 | 0 | 0 | $\dots$ | + | 0. | 3 | 3 | 3 | 3 | 3 | 3 | 3 | $\dots$ | + | 0. | 1 | 4 | 2 | 5 | 9 | 2 | 6 | $\dots$ | + | 0. | 9 | 9 | 9 | 9 | 8 | 9 | 7 | $\dots$ | + | 0. | 2 | 8 | 5 | 1 | 2 | 8 | 3 | $\dots$ | + | 0. | 4 | 2 | 8 | 5 | 1 | 5 | 2 | $\dots$ | + | 0. | 5 | 7 | 2 | 1 | 4 | 2 | 1 | $\dots$ | + | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\ddots$ | + +Em seguida aplicamos sobre esta lista uma função em uma diagonal que altera o valor da entrada em uma unidade, digamos, $f = \{(x,y) : x < 9 \to y = x + 1\ \underline \lor\ x = 9 \to y = 0\}$. + +| 0. | 1 | 1 | 1 | 1 | 1 | 1 | 1 | $\dots$ | +| -------- | -------- | -------- | -------- | -------- | -------- | -------- | -------- | -------- | +| 0. | **2** | 0 | 0 | 0 | 0 | 0 | 0 | $\dots$ | +| 0. | 3 | **4** | 3 | 3 | 3 | 3 | 3 | $\dots$ | +| 0. | 1 | 4 | **3** | 5 | 9 | 2 | 6 | $\dots$ | +| 0. | 9 | 9 | 9 | **0** | 8 | 9 | 7 | $\dots$ | +| 0. | 2 | 8 | 5 | 1 | **3** | 8 | 3 | $\dots$ | +| 0. | 4 | 2 | 8 | 5 | 1 | **6** | 2 | $\dots$ | +| 0. | 5 | 7 | 2 | 1 | 4 | 2 | **2** | $\dots$ | +| $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\ddots$ | + +O número resultante é de forma tal que encontra-se contido nos reais, mas é diferente de todos os infinitos números aqueles com que cruza na tabela, pois difere destes em pelo menos um dígito. Por isso, os números reais são de grandeza superior a uma infinidade contável: estes são incontáveis. + +[^1]: nUSP: 12543033; Turma 04 \ No newline at end of file diff --git a/Matemática Discreta I/Lista 4/ackermann.c b/Matemática Discreta I/Lista 4/ackermann.c new file mode 100644 index 0000000..06211b5 --- /dev/null +++ b/Matemática Discreta I/Lista 4/ackermann.c @@ -0,0 +1,20 @@ +#include + +long long unsigned int ackermann (unsigned int m, unsigned int n) { + if (m == 0) + return n + 1; + if (n == 0) + return ackermann (m - 1, 1); + return ackermann (m - 1, ackermann(m, n - 1)); +} + +int main () { + int m, n; + + printf("Este programa calcula o resultado da função de Ackermann.\nDigite pares de valores inteiros e positivos separados entre si por espaço ou quebra de linha para tê-los calculados, escreva um caractere inválido e pressione ENTER para encerrar:\n"); + + while (scanf(" %d %d", &m, &n) && m >= 0 && n >= 0) + printf("A(%d,%d) = %llu\n", m, n, ackermann(m,n)); + + return 0; +} diff --git a/Matemática Discreta I/Relações e Funções.md b/Matemática Discreta I/Relações e Funções.md deleted file mode 100644 index e1841e0..0000000 --- a/Matemática Discreta I/Relações e Funções.md +++ /dev/null @@ -1,17 +0,0 @@ -# Relações e Funções - -Relações (matemáticas) são relações entre dois ou mais objetos matemáticos (números, funções, etc.) dada alguma **propriedade**. - -**Exemplos**: - -- Números relacionados por desigualdade: $a \le b$, $3 \le 4$, $3 \not \le 2$; - -- Funções relacionadas pelo comportamento assintótico (f ~ O(g)) - -Quando queremos estudar relações entre dois objetos usamos **pares ordenados**. - -$\vdots$ - -## Relações Binárias - -Sejam dois conjuntos $A$ e $B$. Uma relação binária (ou simplesmente *relação*) de $A$ para $B$ é um subconjunto $A \times B : R = \{(a,b)\} \subseteq A \times B$. \ No newline at end of file diff --git a/Matemática Discreta I/Resolução Lista 2 - MD.md b/Matemática Discreta I/Resolução Lista 2 - MD.md deleted file mode 100644 index 6553326..0000000 --- a/Matemática Discreta I/Resolução Lista 2 - MD.md +++ /dev/null @@ -1,148 +0,0 @@ -# Resolução da [Lista 2](https://drive.google.com/file/d/1M3v_yk-E8G9nd8onq83WIpr3zsTuq_iJ/view?usp=drive_web&authuser=0) da disciplina de Matemática Discreta - -> Feita por Guilherme de Abreu Barreto[^1] - -## Funções proposicionais e quantificadores - -### Exercício 1 - -**a)** "Existe pelo menos um elemento no conjunto $A$ tal que somado com 3 fica igual à 10". *Falso*, não há tal elemento. - -**b)** "Todo elemento no conjunto $A$ é tal que somado com 3 fica menor que 10". *Verdadeiro*. - -**c)** "Existe pelo menos um elemento no conjunto $A$ tal que somado com 3 fica menor ou igual a 5". *Verdadeiro*, os elementos 1 e 2. - -**d)** "Existe pelo menos um elemento no conjunto $A$ tal que somado com 3 fica menor que 7". *Verdadeiro*, elementos 1 à 3. - -### Exercício 2 - -**a)** "Existe pelo menos um elemento em $A$ que, para todo elemento em $A$, quando elevado ao cubo possui valor menor que a soma doutro ou do mesmo elemento com 1". *Verdadeiro*, o elemento 1, no caso. - -**b)** "Para todo elemento em $A$ existe um elemento em $A$ cuja soma dos cubos destes elementos é menor que 12". *Verdadeiro*. - -**c)** "Para todo par de elementos em $A$ a soma dos cubos destes é menor que 12". *Falso*. - -### Exercício 3 - -**a)** $(\forall x \in \R)(\exists y \in \R)(\neg P(x,y));$ - -**b)** $(\exists x \in \R)(\exists y \in \R)(\neg P(x,y));$ - -**c)** $(\forall y \in \R) (\forall x \in \R) (\exists z \in \R)(\neg P(x,y)\lor \neg Q(x,z))$ - -### Exercício 4 - -**a)** Existe pelo menos um estudante de SI da EACH que não é do sexo masculino; - -**b)** Nenhum dos estudantes de GPP da EACH tem 25 anos ou mais; - -**c)** Existe pelo menos um estudante da EACH que não mora na ZL. - -### Exercício 5 - -**a)** "Para qualquer número inteiro existe um número inteiro maior que este". *Verdadeiro*. Negação: "Existe um numero inteiro para o qual nenhum número inteiro é maior que este." $(\exist a \in \Z)(\forall b \in \Z)(\neg(a < b))$. - -**b)** "Existe um número inteiro para o qual qualquer número inteiro é menor que ele". *Falso*. Negação: "Para qualquer número inteiro existe pelo menos um número inteiro que não seja menor que ele". $(\forall b \in \Z)(\exists a \in \Z)(\neg(a < b))$. - -### Exercício 6 - -**a)** $(\exists x, y, z \in \Z)P(x,y,z)$ - -**b)** $(n \in \Z^+ : n \le 3)(x_1, \dots , x_n \in \Z)P(x,y,z)$ - -### Exercício 7[^2] - -$$ -(\forall \epsilon \in \R^+)(\exists n \in \N)(\forall n > N)(|x_n - x| < \epsilon) -$$ - -### Exercício 8 - -Proposição: - -$$ -(f, g : \R \to \R) (\forall s \in \R)(\exists r \in \R)(f(r) > 0 \implies g(s) > 0) -$$ - -Negação: - -$$ -(f, g: \R \to \R)(\exists s \in \R)(\forall r \in \R)(\neg(f(r) > 0 \implies g(s) > 0)) -$$ - -A proposição implica que a função $g$ contém $f$, pois esta última influencia o valor da primeira, enquanto o contrário não ocorre. - -## Estratégias de demonstração - -### Exercício 1 - -**Proposição** - -Sejam $n$ e $k$ dois números naturais onde $n > k$. Se tentarmos distribuir $n$ objetos em $k$ urnas ($P$), então pelo menos uma das urnas conterá mais de um objeto ($Q$). $P\implies Q$. - -**Contrapositiva** - -Sejam $n$ e $k$ dois números naturais onde $n > k$. Pelo menos uma urna restará vazia ($\neg Q$) se tentarmos distribuir $k$ objetos em $n$ urnas ($\neg P$). $\neg Q \implies \neg P$. - -### Exercício 2 - -Consideremos $N$ um número de $n$ dígitos $\{a_1, a_2, \dots, a_n\}$ cuja soma $a_1 + a_2 + \dots + a_n = \sum^n_{i = 1} a_i = k$. Sabemos que o critério de divisibilidade de $N$ por 9 é $N | 9 \iff N = 9q$, $\forall N, n, a, k, q \in \N$. Podemos descrever $N$ da seguinte maneira: - -$$ -N = a_1 \cdot 10^{n - 1} + a_2 \cdot 10^{n - 2} + \dots + a_n \cdot 10^{n - n} -= \sum^n_{i = 1}a_i10^{n - i} -$$ - -Por vez, $10^{n}$ pode ser escrito da seguinte forma: - -$$ -10^n = 99\dots9 (\text{n - 1 vezes}) + 1 = \underbrace{\sum^{n - 1}_{i = 1}9\cdot 10^i}_{\equiv\ 9q} + 1 -$$ - -Logo, para cada dígito de $N$ teremos: - -$$ -N = a_1(10^{n - 1} + 1) + a_2(10^{n - 2} + 1) + \dots + a_n (10^{n - n} + 1) = -\\ \sum^n_{i = 1}a_i(9q + 1) = k9q + k -$$ - -Finalmente, $9kq | 9$ e, se $k | 9$ o for, então também é $N | 9$. $\blacksquare$ - -### Exercício 3 - -**Prova direta** - -Se um número n é par, então $n|2 \implies n = 2k$ para qualquer $k \in \R$. - -$$ -n^2 = 2k = \pm \sqrt{2k} \cdot \frac{\sqrt 2}{\sqrt 2} -= 2 \cdot \underbrace{\left(\pm \frac{\sqrt k}{\sqrt 2}\right)}_{k_2} -= 2k_2\ \blacksquare -$$ - -**Prova por contrapositiva** - -Retomemos o enunciado: - -> Se $n^2$ é par ($P$), então $n$ é par ($Q$, $P \implies Q$). - -A contrapositiva disso seria: - -> Se $n$ é ímpar ($\neg Q$), $n^2$ é ímpar ($\neg P$, $\neg Q \implies \neg P$). - -Se $n$ é ímpar, então $n \nmid 2 \implies n = 2k + 1$ para qualquer $k \in \R$. - -$$ -n^2 = n \cdot n = (2k + 1)(2k + 1) = 4k^2 + 2k + 2k + 1 = -2\underbrace{(2k^2 + 2k)}_{k_2} + 1 = 2k_2 + 1\ \blacksquare -$$ - -A primeira demonstração é mais objetiva: foram menores os números de passos requeridos. Mas em termos de compreensibilidade a alternativa possui seu valor. - -### Exercício 4 - - - -[^1]: 12543033 - -[^2]: Resposta retirada diretamente das notas de aula do dia 03/09/2021 \ No newline at end of file diff --git a/Matemática Discreta I/Resolução Lista 3 - MD.md b/Matemática Discreta I/Resolução Lista 3 - MD.md deleted file mode 100644 index 19acc21..0000000 --- a/Matemática Discreta I/Resolução Lista 3 - MD.md +++ /dev/null @@ -1,69 +0,0 @@ -# Resolução da [Lista 2](https://drive.google.com/file/d/1M3v_yk-E8G9nd8onq83WIpr3zsTuq_iJ/view?usp=drive_web&authuser=0) da disciplina de Matemática Discreta - -> Feita por Guilherme de Abreu Barreto[^1] - -## Funções proposicionais e quantificadores - -### Exercício 1 - -**a)** "Existe pelo menos um elemento no conjunto $A$ tal que somado com 3 fica igual à 10". *Falso*, não há tal elemento. - -**b)** "Todo elemento no conjunto $A$ é tal que somado com 3 fica menor que 10". *Verdadeiro*. - -**c)** "Existe pelo menos um elemento no conjunto $A$ tal que somado com 3 fica menor ou igual a 5". *Verdadeiro*, os elementos 1 e 2. - -**d)** "Existe pelo menos um elemento no conjunto $A$ tal que somado com 3 fica menor que 7". *Verdadeiro*, elementos 1 à 3. - -### Exercício 2 - -**a)** "Existe pelo menos um elemento em $A$ que, para todo elemento em $A$, quando elevado ao cubo possui valor menor que a soma doutro ou do mesmo elemento com 1". *Verdadeiro*, o elemento 1, no caso. - -**b)** "Para todo elemento em $A$ existe um elemento em $A$ cuja soma dos cubos destes elementos é menor que 12". *Verdadeiro*. - -**c)** "Para todo par de elementos em $A$ a soma dos cubos destes é menor que 12". *Falso*. - -### Exercício 3 - -**a)** $(\forall x \in \R)(\exists y \in \R)(\neg P(x,y));$ - -**b)** $(\exists x \in \R)(\exists y \in \R)(\neg P(x,y));$ - -**c)** $(\forall y \in \R) (\forall x \in \R) (\exists z \in \R)(\neg P(x,y)\lor \neg Q(x,z))$ - -### Exercício 4 - -**a)** Existe pelo menos um estudante de SI da EACH que não é do sexo masculino; - -**b)** Nenhum dos estudantes de GPP da EACH tem 25 anos ou mais; - -**c)** Existe pelo menos um estudante da EACH que não mora na ZL. - -### Exercício 5 - -**a)** "Para qualquer número inteiro existe um número inteiro maior que este". *Verdadeiro*. Negação: "Existe um numero inteiro para o qual nenhum número inteiro é maior que este." $(\exist a \in \Z)(\forall b \in \Z)(\neg(a < b))$. - -**b)** "Existe um número inteiro para o qual qualquer número inteiro é menor que ele". *Falso*. Negação: "Para qualquer número inteiro existe pelo menos um número inteiro que não seja menor que ele". $(\forall b \in \Z)(\exists a \in \Z)(\neg(a < b))$. - -### Exercício 6 - -**a)** $(\exists x, y, z \in \Z)P(x,y,z)$ - -**b)** $(n \in \Z^+ : n \le 3)(x_1, \dots , x_n \in \Z)P(x,y,z)$ - -### Exercício 7 - -$$ -(\forall \epsilon \in \R^+)(\exists n \in \N)(\forall n > N)(|x_n - x| < \epsilon) -$$ - -### Exercício 8 - -Proposição: - -$$ -(f, g : \R \to \R) (\forall s \in \R)(\exists r \in \R)(f(r) > 0 \implies g(s) > 0) -$$ - -## Estratégias de demonstração - -[^1]: 12543033 \ No newline at end of file diff --git a/Matemática Discreta I/Álgebra Booleana.md b/Matemática Discreta I/Álgebra Booleana.md new file mode 100644 index 0000000..e5aa242 --- /dev/null +++ b/Matemática Discreta I/Álgebra Booleana.md @@ -0,0 +1,182 @@ +# Algebra booleana + +## Relação com outras algebras + +É possível constatar uma semelhança entre as operações encontradas nas algebras de proposição e de conjuntos, de tal forma a encontrarmos as seguintes equivalências: + +- $V \equiv \Omega$ + +- $\varnothing \equiv F$ + +- $\cup \equiv \lor$ + +- $\cap \equiv \land$ + +Isso se deve ao fato destas serem instâncias de um mesmo tipo de algebra: a **algebra booleana**. + +## Definição de algebra booleana + +A algebra booleana está definida para os seguintes termos: + +- Existe um conjunto $S$ contendo pelo menos dois elementos ditos **especiais**, aqui denominados 0 e 1, onde + + - 0 é o elemento nulo aditivo; + + - e 1 é o elemento nulo multiplicativo. + +- Existem duas operações binárias (que relacionam dois elementos): + + - Adição ($+$) e + + - Multiplicação ($\cdot$)[^1]; + +- Existe uma operação unária que associa cada elemento $x \in S$ a um elemento $x' \in S$, o complemento de $x$ em $S$. + +### Axiomas + +A sêxtupla $B$ de elementos $B = \lang S, 0, 1, + , \cdot, ' \rang$ é uma algebra booleana se esta exibe as seguintes propriedades: + +**P1.** Comutatividade + +- $x + y = y + x$ + +- $x \cdot y = y \cdot x$ + +**P2.** Associatividade + +- $(x + y) + z = x + (y + z)$ + +- $(x \cdot y)\cdot z = x \cdot (y \cdot z)$ + +**P3.** Distributividade + +- $x + (y \cdot z) = (x + y) \cdot (x + z)$[^2] + +- $x \cdot (y + z) = (x \cdot y) + (x \cdot z)$ + +**P4.** Identidade + +- $x + 0 = x$ + +- $x \cdot 1 = x$ + +**P5.** Complementariedade + +- $x + x' = 1$ + +- $x \cdot x' = 0$ + +**P6.** Precedência: $' > \cdot > +$. Exemplo: + +- $x \cdot y + z = (x \cdot y) + z$ + +- $x \cdot y' = x \cdot(y')$ + +Por isso a relação com as algebras de proposição e de conjuntos: estas respeitam esses axiomas e podem ser descritas em termos das sextuplas + +- $B = \lang P(A), \varnothing, A, \cup, \cap,' \rang$ (Algebra de conjuntos); + +- $B = \lang \mathfrak L, F, V, \lor, \land, \neg \rang$ (Algebra de proposição); + +- $B = \lang \{0,1\}, 0, 1, ||, \&\&, ! \rang$ (Algebra binária). + +### Dualidade + +Tal qual demonstrado nos axiomas, cada propriedade **dual** obtida pelas substituições $+ \leftrightarrow \cdot$, $0 \leftrightarrow 1$ também é válida. Assim como a reflexividade das relações de igualdade como, por exemplo, + +$$ +x \cdot (y + z) = (x \cdot y) + (x \cdot z) \leftrightarrow +(x \cdot y) + (x \cdot z) = x \cdot (y + z) +$$ + +### Resultados + +**R1.** Em qualquer algebra booleana todo elemento $x$ satisfaz: + +- $x + x = x$ + **Prova:** + $x = x + 0 = x + xx' = (x + x)(x + x') = (x + x)\cdot 1 = x + x$ + +- $xx = x$ + **Prova:** é a expressão dual de $x + x = x$ + +**R2. Lei da absorção:** em qualquer algebra booleana quaisquer elementos $x,y$ satisfazem + +- $x + xy = x$ + **Prova:** faremos uma analogia com a algebra proposicional e analisaremos a tabela verdade destas proposições: + + | x | y | xy | x + xy | + |:---:|:---:|:---:|:------:| + | F | F | F | F | + | F | V | F | F | + | V | F | F | V | + | V | V | V | V | + +- $x (x + y) = x$ + **Prova:** é a expressão dual de $x + xy = x$. + +**R3. Lei da nulidade:** em qualquer algebra booleana todo elemento $x$ satisfaz + +- $x + 1 = 1$ + + **Prova:** + $x + 1 = x + \underbrace{(x + x')}_{\textbf{P5}} = \underbrace{(x + x)}_{\textbf{P2}} + x' = \underbrace{x}_{\textbf{R1}} + x' = \underbrace{1}_{\textbf{P5}}$ + +- $0x = 0$ + **Prova:** é a expressão dual de $x + 1 = 1$. + +**R4.** Em qualquer algebra booleana quaisquer elementos $x, y$ tais que $x + y = 1$ e $xy = 0$, então $x = y'$. + +**Prova:** Conforme a propriedade **P5**, $y + y' = 1$ e $yy' = 0$. Então se $x = y'$, tem-se que $x + y = 1$ e $xy = 0$. + +**R5.** Em qualquer algebra booleana todo elemento $x$ satisfaz $x'' = (x')' = x$. Também temos que $0' = 1$ e portanto $1' = 0$. + +**R6. Leis de DeMorgan:** Em qualquer algebra booleana quaisquer elementos $x$, $y$ satisfazem: + +- $(x + y)' = x'y'$ + +- $(xy)' = x' + y'$ + +## Expressões Booleanas + +Uma **expressão booleana** nas variáveis $x, y, \dots$ em uma algebra booleana $B$ é uma expressão bem formada pelas **variáveis** muitas vezes em conjunção com **operadores** e, por vezes, **parênteses**. Por exemplo: + +$$ +\alpha(x,y,z) = xyz' + x(y + z) + (x + z)' +$$ + +é uma expressão booleana pois podemos calcular seu valor nos elementos 0 e 1 (que pertencem a toda álgebra booleana): + +$$ +\alpha(0,1,0) = 0\cdot1 \cdot 0' + 0(1 + 0) + (0 + 0)' = 0 + 0 + 1 = 1 +$$ + +**Teorema:** Toda expressão booleana pode ser colocada na forma de uma **soma de produtos fundamentais**, expressões que não contêm variáveis repetidas ou valem 0. Exemplos: + +- $xy'z$ é um produto fundamental; + +- $0 \cdot0\cdot\dots\cdot0$ é um produto fundamental; + +- $xytx'$ não é (x está repetido). Posto em forma de produto fundamental, $xytx' = 0$. + +### Variáveis e literais + +Variáveis são os valores aqueles denominados por incógnitas $x, y, z$, etc. literais são instâncias destas variáveis na expressão. Por exemplo, em $xytx'$ temos duas literais, $x$ e $x'$, da variável $x$. + +### Inclusão + +Se $\alpha$ e $\beta$ são duas expressões booleanas, dizemos que $\beta$ inclui $\alpha$, denotado por $\alpha \le \beta$, se todo literal de $\alpha$ também aparece em $\beta$. Por exemplo: $xz' \le xyz't$, mas $xz \not \le xyz't$. + +**Teorema:** pela lei da absorção, se $\alpha \le \beta$, então $\alpha + \beta = \alpha$. Exemplo: + +$\begin{cases} +\alpha(x,z) = xz \\ +\beta(x,y,z) = xyz +\end{cases}\\ +\therefore \alpha + \beta = xz + xyz = xz + (xz)y = xz$ + +### Formas disjuntivas + +[^1]: Usualmente este sinal é omitido. + +[^2]: Note a distinção com a algebra artimética e a semelhança com a algebra de conjuntos. \ No newline at end of file

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b/Introdução à Análise de Algoritmos/Lista 1/Resolução Lista 1 - IAA.md similarity index 71% rename from Introdução à Análise de Algoritmos/Resolução Lista 1 - IAA.md rename to Introdução à Análise de Algoritmos/Lista 1/Resolução Lista 1 - IAA.md index 9437527..48a54eb 100644 --- a/Introdução à Análise de Algoritmos/Resolução Lista 1 - IAA.md +++ b/Introdução à Análise de Algoritmos/Lista 1/Resolução Lista 1 - IAA.md @@ -6,24 +6,30 @@ ```c m ← 0 // c_2 -for i ← 1 até n // c_2 + (n - 1) c_1 - do for j ← 1 até n // c_2 + (n - 1)² c_1 - do for k ← 1 até n // c_2 + (n - 1)³ c_1 +for i ← 1 até n // c_2 + c_5 + (n - 1) c_1 + do for j ← 1 até n // c_2 + c_5 + (n - 1)² c_1 + do for k ← 1 até n // c_2 + c_5 + (n - 1)³ c_1 if a_i + b_j + c_k = 0 // n³ (2 c_3 + c_5) - then m ← m + 1 // 0 c_1 até n³ c_1 + then m ← m + 1 // 0 (c_2 + c_3) até n³ (c_2 + c_3) return m // c_4 ``` -Assim o sendo, temos que este algoritmo, na ausência de valores capazes de satisfazer a condição de iteração da variável $m$ (melhor caso) tem um tempo de execução equivalente à: +> **Obs:** Admite-se $$ -(n^3 - 2n^2 + 2n - 1)c_1 + 4c_2 + 2n^3c_3 + c_4 + n^3c_5 +c_1 (\text{iteração})= c_3 (\text{soma}) + c_2 (\text{atribuição}) + c_5 (\text{comparação ‒ com o valor }n) $$ -Enquanto quando este encontra valores compatíveis, este tem um tempo de execução equivalente à: +Assim o sendo, temos que este algoritmo, na ausência de valores capazes de satisfazer a condição de acréscimo da variável $m$ (melhor caso) tem um tempo de execução equivalente à: $$ -[2(n^3 - n^2 + n) - 1]c_1 + 4c_2 + 2n^3c_3 + c_4 + n^3c_5\ \blacksquare +(n^3 - 2n^2 + n - 1)c_1 + 4c_2 + 2n^3 c_3 + c_4 + (n^3 + 3)c_5 +$$ + +Enquanto quando este encontra apenas valores compatíveis (pior caso), este tem um tempo de execução equivalente à: + +$$ +(n^3 - 2n^2 + n - 1)c_1 + (n^3 + 4)c_2 + 3n^3 c_3 + c_4 + (n^3 + 3)c_5\ \blacksquare $$ ## Exercício 2 @@ -40,20 +46,21 @@ $$ 0 \le c_1g(n) \le f(n) \le c_2g(n) $$ -Assim sendo, temos que em nosso caso $g(n) = n^3$, enquanto$f(n) = 4n^3 + n^2 + n + 2$ (a função equivalente ao pior caso, igualando as constantes à 1) Para qualquer $n \ge 1$. Assim sendo, tem-se: +Assim sendo, temos que em nosso caso $g(n) = n^3$, enquanto a função equivalente ao pior caso, igualando as constantes à 1, e substituindo $c_1$ por $c_2 + c_3 + c_5$ é $f(n) = 8n^3 - 6 n^2 + 3n + 5$, para qualquer $n \ge 1$. Assim sendo, tem-se: $$ -0 \le c_1n^3 \le 4n^3 + n^2 + n + 2 \le c_2n^3\\\ \\ -0 \le c_1 \le 4 + \dfrac 1n + \dfrac 1{n^2} + \dfrac 2{n^3} \le c_2 \implies 0 \le c_1 \le 4\ e\ c_2 \ge 8 +0 \le c_1n^3 \le 8n^3 - 6n^2 + 3n + 5 \le c_2n^3 $$ +Utilizando-se de uma calculadora eletrônica, obtemos que $0 \le c_1 ⪅ 2.16$ e $10 \le c_2$. + De fato, vemos que este resultado se concretiza para valores de $n \ge 1$: -![](/home/user/Public/USP/Sistemas%20de%20Informação/2º%20semestre/Introdução%20à%20Análise%20de%20Algoritmos/Imagens/2021-10-01-22-55-58-image.png) + -> Linha verde: $f(n)$; linha cinza: $c_1g(n)$; e linha rosa: $c_2g(n)$ +> Linha azul: $f(n)$; linha verde: $2.16\,g(n)$; e linha vermelha: $10\,g(n)$ -Portanto, o tempo de processamento do algorítimo 3Soma é $\Theta(n^3)$ no pior caso. $\blacksquare$ +Portanto, o tempo de processamento do algoritmo 3Soma é $\Theta(n^3)$ no pior caso. $\blacksquare$ ## Exercício 3 @@ -87,8 +94,8 @@ end m ← 0 insertionSort (C, sizeof(C)) - for i ← 1 to n - do for j ← 1 to n + for i ← 1 to sizeof(A) + do for j ← 1 to sizeof(B) do if binarySearch (C, sizeof(C), -(A[i] + B[j])) then m ← m + 1 return m @@ -115,7 +122,7 @@ $$ T(n) \in \Theta(n^2\log n) $$ -Sendo esta a linha com maior número de iterações no algoritmo `3Soma`, podemos, por extensão, afirmar que a notação $\Theta$ anteriormente descrita é representativa do tempo de execução do algoritmo como um todo. $\blacksquare$ +Sendo esta a linha com maior número de iterações no algoritmo `3Soma`, podemos, por extensão, afirmar que a notação $\Theta$ anteriormente descrita é representativa do tempo de execução do algoritmo como um todo. $\blacksquare$ ## Exercício 4 @@ -145,4 +152,4 @@ Ao longo de sua execução, é invariável que $m$ equivale ao número de somas RIBEIRO, M. **Introdução à Análise de Algoritmos**. Disponível em: . 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zcP}j|A(&s|fWMxnc#>!={-+QVwFs{Wk5m>z500Xfh!>O_B2Mn7fRR8di;zdiW00Uy zvcUE7f>`bkiW)}YSfirmLUJJrD5OwSG!+;GCyr`e#>bY0iE)eK ZmYUhlDtf;Hd`}ySs-l)cnY>xh{{y5M5Ay&3 literal 0 HcmV?d00001 diff --git a/Matemática Discreta I/Lista 2/Resolução Lista 2 - MD.md b/Matemática Discreta I/Lista 2/Resolução Lista 2 - MD.md new file mode 100644 index 0000000..1bca729 --- /dev/null +++ b/Matemática Discreta I/Lista 2/Resolução Lista 2 - MD.md @@ -0,0 +1,605 @@ +# Resolução da [Lista 2](https://drive.google.com/file/d/1M3v_yk-E8G9nd8onq83WIpr3zsTuq_iJ/view?usp=drive_web&authuser=0) da disciplina de Matemática Discreta + +> Feita por Guilherme de Abreu Barreto[^1] + +## Funções proposicionais e quantificadores + +### Exercício 1 + +**a)** "Existe pelo menos um elemento no conjunto $A$ tal que somado com 3 fica igual à 10". *Falso*, não há tal elemento. + +**b)** "Todo elemento no conjunto $A$ é tal que somado com 3 fica menor que 10". *Verdadeiro*. + +**c)** "Existe pelo menos um elemento no conjunto $A$ tal que somado com 3 fica menor ou igual a 5". *Verdadeiro*, os elementos 1 e 2. + +**d)** "Todo elemento no conjunto $A$ é tal que somado com 3 fica menor que 7". *Falso*. + +### Exercício 2 + +**a)** "Existe pelo menos um elemento em $A$ que, para todo elemento em $A$, quando elevado ao quadrado possui valor menor que a soma doutro ou do mesmo elemento com 1". *Verdadeiro*, o elemento 1, no caso. + +**b)** "Para todo elemento em $A$ existe um elemento em $A$ cuja soma dos quadrados destes elementos é menor que 12". *Verdadeiro*. + +**c)** "Para todo par de elementos em $A$ a soma dos quadrados destes é menor que 12". *Falso*. + +### Exercício 3 + +**a)** $(\forall x \in \R)(\exists y \in \R)(\neg P(x,y));$ + +**b)** $(\exists x \in \R)(\exists y \in \R)(\neg P(x,y));$ + +**c)** $(\forall y \in \R) (\forall x \in \R) (\exists z \in \R)(\neg P(x,y)\lor \neg Q(x,z))$ + +### Exercício 4 + +**a)** Existe pelo menos um estudante de SI da EACH que não é do sexo masculino; + +**b)** Nenhum dos estudantes de GPP da EACH tem 25 anos ou mais; + +**c)** Existe pelo menos um estudante da EACH que não mora na ZL. + +### Exercício 5 + +**a)** "Para qualquer número inteiro existe um número inteiro maior que este". *Verdadeiro*. Negação: "Existe um numero inteiro para o qual nenhum número inteiro é maior que este." $(\exist a \in \Z)(\forall b \in \Z)(\neg(a < b))$. + +**b)** "Existe um número inteiro para o qual qualquer número inteiro é menor que ele". *Falso*. Negação: "Para qualquer número inteiro existe pelo menos um número inteiro que não seja menor que ele". $(\forall b \in \Z)(\exists a \in \Z)(\neg(a < b))$. + +### Exercício 6 + +**a)** $(\exists x, y, z \in \Z)P(x,y,z)$ + +**b)** $(x \in \Z)(n \in \N : n \le 3)P(x_1, \dots, x_n)$ + +### Exercício 7[^2] + +$$ +(\forall \epsilon \in \R^+)(\exists n \in \N)(\forall n > N)(|x_n - x| < \epsilon) +$$ + +### Exercício 8 + +Proposição: + +$$ +(f, g : \R \to \R) (\forall s \in \R)(\exists r \in \R)(f(r) > 0 \implies g(s) > 0) +$$ + +Negação: + +$$ +(f, g: \R \to \R)(\exists s \in \R)(\forall r \in \R)(\neg(f(r) > 0 \implies g(s) > 0)) +$$ + +A proposição implica que a função $g$ contém $f$, pois esta última influencia o valor da primeira, enquanto o contrário não ocorre. + +## Estratégias de demonstração + +### Exercício 1 + +**Proposição** + +Sejam $n$ e $k$ dois números naturais onde $n > k$. Se tentarmos distribuir $n$ objetos em $k$ urnas ($P$), então pelo menos uma das urnas conterá mais de um objeto ($Q$). $P\implies Q$. + +**Contrapositiva** + +Sejam $n$ e $k$ dois números naturais onde $n > k$. Pelo menos uma urna restará vazia ($\neg Q$) se tentarmos distribuir $k$ objetos em $n$ urnas ($\neg P$). $\neg Q \implies \neg P$. + +### Exercício 2 + +Consideremos $N$ um número de $n$ dígitos $\{a_1, a_2, \dots, a_n\}$ cuja soma $a_1 + a_2 + \dots + a_n = \sum^n_{i = 1} a_i = k$. Sabemos que o critério de divisibilidade de $N$ por 9 é $N | 9 \iff N = 9q$, $\forall N, n, a, k, q \in \N$. Podemos descrever $N$ da seguinte maneira: + +$$ +N = a_1 \cdot 10^{n - 1} + a_2 \cdot 10^{n - 2} + \dots + a_n \cdot 10^{n - n} += \sum^n_{i = 1}a_i10^{n - i} +$$ + +Por vez, $10^{n}$ pode ser escrito da seguinte forma: + +$$ +10^n = 99\dots9 (\text{n - 1 vezes}) + 1 = \underbrace{\sum^{n - 1}_{i = 1}9\cdot 10^i}_{\equiv\ 9q} + 1 +$$ + +Logo, para cada dígito de $N$ teremos: + +$$ +N = a_1[(10^{n - 1}- 1) + 1] + a_2((10^{n - 2} - 1) + 1] + \dots ++ a_n [(10^{n - n} - 1) + 1] = +\\ \sum^n_{i = 1}a_i(9q + 1) = k9q + k +$$ + +Finalmente, $9kq | 9$ e, se $k | 9$ o for, então também é $N | 9$. $\blacksquare$ + +### Exercício 3 + +**Prova direta** + +Se um número n é par, então $n|2 \implies n = 2k$ para qualquer $k \in \R$. + +$$ +n^2 = 2k = \pm \sqrt{2k} \cdot \frac{\sqrt 2}{\sqrt 2} += 2 \cdot \underbrace{\left(\pm \frac{\sqrt k}{\sqrt 2}\right)}_{k_2} += 2k_2\ \blacksquare +$$ + +**Prova por contrapositiva** + +Retomemos o enunciado: + +> Se $n^2$ é par ($P$), então $n$ é par ($Q$, $P \implies Q$). + +A contrapositiva disso seria: + +> Se $n$ é ímpar ($\neg Q$), $n^2$ é ímpar ($\neg P$, $\neg Q \implies \neg P$). + +Se $n$ é ímpar, então $n \nmid 2 \implies n = 2k + 1$ para qualquer $k \in \R$. + +$$ +n^2 = n \cdot n = (2k + 1)(2k + 1) = 4k^2 + 2k + 2k + 1 = +2\underbrace{(2k^2 + 2k)}_{k_2} + 1 = 2k_2 + 1\ \blacksquare +$$ + +A primeira demonstração é mais objetiva: foram menores os números de passos requeridos. Mas em termos de compreensibilidade a alternativa possui seu valor. + +### Exercício 4 + +Um número primo inteiro, $p \in \Z$ é aquele que tem **somente** quatro divisores distintos, $\pm 1$ e $\pm p$. Já um número primo natural, $p \in \N$ tem **unicamente** dois divisores naturais distintos: o número um e ele mesmo. Por estarmos tratando aqui de valores para $p$ tais que $p > 3$, estamos tratando de números primos **naturais**. Seria a fórmula $3k \pm 1$, $k \in \N$, capaz de representá-los? + +- Todo número $p > 2$ é **ímpar**, doutra forma seria divisível por dois e não primo. Se segue que todo número p ímpar maior que 3 **não é divisível** por 3. + +- Múltiplos de três são ora ímpar, ora par: + + - $3(2k) = 2(3k) \equiv 0 (\text{mod }2)$, par; + + - $3 (2k \pm 1) = 6n \pm 3 = 6k \pm 2 \pm 1 = 2(3k \pm 1) \pm 1 \equiv 0(\text{mod }2) \pm 1$, ímpar. + +Assim, proponho que esta fórmula seja capaz de representar todos os números naturais impares $n_i$ não múltiplos de 3 para todo valor $2k$. + +$n_i = 2k \pm 1 \equiv 0 (\text{mod }2) \pm 1\\ +3(2k) \pm 1 = 2(3k) \pm 1 \equiv 0 (\text{mod }2) \pm 1\\ +\therefore n_i \equiv 3(2k) \pm 1\ \blacksquare +$ + +### Exercício 5 + +Sempre que qualquer um dos lados de uma inequação sofre uma multiplicação por um valor menor que 0, o sinal de desigualdade assume sua forma dual, de tal sorte que $-2 < 1$ ao ser elevado ao quadrado fica: + +$$ +(-2)\underbrace{(-2)}_{<\ 0} > 1 \cdot 1 \implies 4 > 1\ \blacksquare +$$ + +### Exercício 6 + +**a.** Conforme exposto anteriormente, qualquer número ímpar pode ser representado pela forma $2k \pm 1 \equiv 0 (\text{mod } 2) \pm 1$, utilizaremo-nos aqui da forma $n = 2k + 1$: + +$n^2 + 4n = (2k + 1)^2 + 4(2k + 1) = 4k^2 + 4k + 1 + 8k + 4 =\\ 2(2k^2 + 2k + 2) + 1 \equiv 0 (\text{mod }2) + 1 \text{; é ímpar.}\ \blacksquare$ + +**b.** A contrapositiva dessa afirmação é: se $r$ é racional então $r^2$ é racional. Por definição um número racional é aquele que pode ser representado na forma + +$$ +\frac n d, n \in \Z, d \in \Z^* +$$ + +Então, + +$$ +r = \frac nd \implies r^2 = \frac{n^2}{d^2} = \dfrac{n_2}{d_2}, n_2 \in \Z, d \in \Z^*\ \blacksquare +$$ + +**c.** Falso. Por exemplo, para $n = 2$ este não é o caso: + +$$ +2^5 < 5^n \implies 32 < 25 \equiv F +$$ + +Enquanto para $n = 1$ é: + +$$ +1^5 < 5^1 +$$ + +Então seria correto dizer que $(\exists n \in \N)(n^5 < 5^n)$. $\blacksquare$ + +**d.** Na matemática, considera-se triviais soluções ou exemplos ridiculamente simples e de pouco interesse. Muitas vezes, as soluções ou exemplos triviais que envolvem o número 0 são considerados triviais. Este não é o caso com a desigualdade de Bernoulli, que tem implicações importantes para a análise combinatória necessita ser demonstrada por indução finita. Isto é, admite-se 0 enquanto base de indução, mas então procede-se a demonstrar que tal hipótese vale para qualquer número natural $n$. Tal demonstração se dá da seguinte maneira: + +1. $(1 + r)^n \ge 1 + nr$ é válido para $n = 0$: $(1+r)^0 = 1; 1 \ge 1 + 0r$. + +2. Agora, veremos se isso é válido para $n + 1$: + +$(1 + r)^n \ge 1 + nr \implies (1 + r)(1 + r)^n \ge (1 + r)(1 + nr) \implies \\ (1 + r)^{n + 1} \ge 1 + nr + r + \underbrace{nr^2}_{\ge\ 0}$ + +Repare que em $(1 + r)^{n + 1} \ge 1 + nr + r + nr^2$, $nr^2$ é sempre maior ou igual à 0, então + +$(1 + r)^{n + 1} \ge 1 + nr + r \implies (1 + r)^{n + 1} \ge 1 + r(n + 1)\ \blacksquare$ + +Fica demonstrado que tal igualdade é válida para qualquer $n \in \N$. + +### Exercício 7 + +**a.** Considerando a progressão aritmética $1, \dots, n$, a soma de todos os termos desta progressão ($S_n$)pode ser escrita das seguintes formas: + +$$ +S_n = 1 + \dots + n \\ +S_n = n + \dots + 1 +$$ + +Somando estas formulações membro a membro, obtemos: + +$$ +2 S_n = (1 + n) + (2 + (n - 1)) + \dots + ((n - 1) + 2) + (n + 1) +$$ + +Nesta formulação, notemos que + +- Todos os pares entre parênteses têm o mesmo valor, por serem simétricos em relação às extremidades da progressão; + +- existem n pares. + +Logo, + +$$ +2 S_n = n(1 + n) \implies S_n = \frac{n(n + 1)}2 +$$ + +Consideremos agora a soma dos cubos $1^3, \dots , n^3$. O produto notável cubo da soma pode ser descrito da seguinte forma: + +$$ +(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 +$$ + +Então, enquanto $k^3 = k^3$ para qualquer número $k \in \Z$, temos que + +$$ +(k + 1)^3 = k^3 + 3k^2 + 3k + 1 +$$ + +Então, $1^3 + \dots + n^3$ equivale à + +$$ ++ \begin{cases} +\textcolor{red}{\cancel{1^3}} = 1^3 \\ +\textcolor{magenta}{\cancel{2^3}} = 1^3 + 3(1)^2 + 3(1) + \textcolor{red}{\cancel 1} \\ +\textcolor{blue}{\cancel{3^3}} = \textcolor{magenta}{\cancel{2^3}} + 3(2)^2 + 3(2) + 1 \\ +\cancel{4^3} = \textcolor{blue}{\cancel{3^3}} + 3(3)^2 + 3(3) + 1 \\ +\vdots \\ +\cancel{n^3} = \cancel{(n - 1)^3} + 3(n-1)^2 + 3(n - 1) + 1 \\ +(n + 1)^3 = \cancel{n^3} + 3n^2 + 3n + 1 +\end{cases} \\ \implies +(n + 1)^3 = 1^3 + 3 S_{n^2} + 3 S_n + (n + 1) +$$ + +Onde $S_{n^2}$ é o valor da soma dos quadrados de $1, \dots, n$, ou seja, o valor que buscamos. Resolvendo essa equação, temos: + +$(n + 1)^3 = 1 + 3S_{n^2} + 3 \dfrac{n(n + 1)}2 + (n + 1)\\ +\implies 6 S_{n^2} = 2 (n + 1)^3 - 3n(n+ 1) - 2(n + 1)\\ = (n + 1)[2(n+ 1)^2 - 3n - 2]\\ = (n + 1)[2(n^2 + 2n + 1) - 3n - 2]\\ = (n + 1) (2n^2 + 4n + \cancel 2 - 3n - \cancel 2)\\ = n(n + 1)(2n + 1)\\ \implies S_{n^2} = \dfrac{n(n + 1)(2n + 1)}6\ \blacksquare$ + +**b.** + +1. A hipótese se conforma na base de indução: $1^3 = 1^2$. + +2. Se assumirmos que + +$$ +1^3 + \dots + n^3 = \left[\dfrac{n(n + 1)}2\right]^2 +$$ + +é verdadeiro, então + +$$ +1^3 + \dots + (n + 1)^3 = \left[\dfrac{(n + 1)(n + 2)}2\right]^2 +$$ + +também o é. + +3. Testemos esta hipótese de indução: + +$1^3 + \dots + n^3 + (n + 1)^3 = \left[\dfrac{n(n + 1)}2\right]^2 + (n + 1)^3 = (n + 1)^2 \left(\dfrac{n^2}4 + n + 1\right)\\ = \dfrac{(n + 1)^2(n^2 + 4n + 4)}4 = \dfrac{(n + 1)^2(n + 2)^2}4 = \left[\dfrac{(n + 1)(n + 2)}2\right]^2\ \blacksquare$ + +**c.** + +- Base de indução ($n = 2$): + +$\displaystyle \frac{a_1}{1 + a_1} + \frac{a_2}{(1 + a_1) (1 + a_1 + a_2)} = \frac{a_1(1 + a_1 + a_2) + a_2}{(1 + a_1) (1 + a_1 + a_2)} \\\ \\ = \frac{a_1 + a_1^2 + a_1a_2 + a_2}{(1 + a_1) (1 + a_1 + a_2)} = \frac{a_1\ \cancel{(1 + a_1)} + a_2\ \cancel{(1 + a_1)}}{\cancel{(1 + a_1)}(1 + a_1 + a_2)} = \frac{a_1 + a_2}{1 + a_1 + a_2}$ + +- Hipótese de indução: se + +$$ +\frac{a_1 + \dots + a_n}{1 + a_1 + \dots a_n} + = \frac{a_1}{1 + a_1} + \dots + +\frac{a_n}{(1 + a_1 + \dots + a_{n - 1})(1 + a_1 + \dots + a_n)} +$$ + +para todo $n \in \N$, então + +$$ +\frac{a_1 + \dots + a_{n + 1}}{1 + a_1 + \dots a_{n + 1}} + = \frac{a_1}{1 + a_1} + \dots + +\frac{a_{n + 1}}{(1 + a_1 + \dots + a_n)(1 + a_1 + \dots + a_{n + 1})} +$$ + +- Passo de indução: + +$\displaystyle \frac{a_1}{1 + a_1} + \dots + +\frac{a_{n + 1}}{(1 + a_1 + \dots + a_n)(1 + a_1 + \dots + a_{n + 1})} =\\\ \\ \underbrace{\frac{a_1 + \dots + a_n}{1 + a_1 + \dots a_n}}_{\text{Por hipótese}} + \frac{a_{n + 1}}{(1 + a_1 + \dots + a_n)(1 + a_1 + \dots + a_{n + 1})} +$ + +Prosseguimos com a substituição de variáveis $a_1 + \dots + a_n = A$: + +$\displaystyle \frac{A}{1 + A} + \frac{a_{n + 1}}{(1 + A) (1 + A + a_{n + 1})} = \frac{A(1 + A + a_{n + 1}) + a_{n + 1}}{(1 + A) (1 + A + a_{n + 1})} \\\ \\ = \frac{A + A^2 + Aa_{n + 1} + a_{n + 1}}{(1 + A) (1 + A + a_{n + 1})} = \frac{A\ \cancel{(1 + A)} + a_{n + 1}\ \cancel{(1 + A)}}{\cancel{(1 + A)}(1 + A + a_{n + 1})} = \frac{a_1 + \dots + a_{n + 1}}{1 + a_1 + \dots a_{n + 1}}\,\blacksquare$ + +### Exercício 8 + +**a.** Três é um número ímpar e, portanto, pode ser escrito na forma $(2k + 1)$. Logo, + +$(2k + 1)^n - 1 = (2^nk^n + 2^{n - 1}k^{n - 1} + \dots + 2k + \cancel 1) - \cancel 1\\ = 2(2^{n - 1}k^n + 2^{n - 2}k^{n - 1} + \dots + k) \equiv 0(\text{mod 2})$ + +Ou seja, um número par. $\blacksquare$ + +**b.** Se $n$ é par, então + +$5^n - 2^n = 5^{2k} - 2^{2k} = (5^k - 2^k) (5^k + 2^k)\\ = (5 - 2)(5^{k - 1} + 5^{k - 2} \cdot 2 + \dots + 5 \cdot 2^{k - 2} + 2^{k - 1})(5^k + 2^k) \\ = 3(5^{k - 1} + 5^{k - 2} \cdot 2 + \dots + 5 \cdot 2^{k - 2} + 2^{k - 1})(5^k + 2^k) \equiv 0 (\text{mod 3})$ + +Senão, + +$5^{2k +1} - 2^{2k + 1} = (5 - 2)(5^{2k} + 5^{2k - 1} \cdot 2 + \dots + 5 \cdot 2^{2k - 1} + 2^{2k}) = \\ 3(5^{2k} + 5^{2k - 1} \cdot 2 + \dots + 5 \cdot 2^{2k - 1} + 2^k) \equiv 0 (\text{mod 3})\ \blacksquare$ + +**c.** $2^n + 3^n$ é múltiplo de 5 quando $n = 2k + 1$ (ida): + +$2^{2k +1} - 3^{2k + 1} = (2 + 3)(2^{2k} + 2^{2k - 1} \cdot 3 + \dots + 2 \cdot 3^{2k - 1} + 3^k) = \\ 5(2^{2k} + 2^{2k - 1} \cdot 3 + \dots + 2 \cdot 3^{2k - 1} + 3^k) \equiv 0 (\text{mod 5})$ + +o número 5, ao ser multiplicado, **não** produz um número da forma $2^n + 3^n$, quando $n = 2k$ (volta): + +- Toda potência de 5 tem como último algarismo 5, pois o resultado da multiplicação de 5 por 5 é 25. + +- Toda potência $2^{2k} = 4^k$, $4 \times 4 = 1\textbf 6; 16 \times 4 = 2\textbf 4; 24 \times 4 = 9\textbf 6$, e assim por diante. Ou seja, ora se produz final 6, ora se produz final 4. + +- Toda potência $3^{2k} = 9^k, 9 \times 9 = 8\textbf 1; 81 \times 9 = 72\textbf 9; 729 \times 9 = 656\textbf 1$, e assim por diante. Ou seja, ora se produz final 1, ora se produz final 9. + +Como 1 + 6 = 7 produz final 7 e 9 + 4 = 13 produz final 3, não é possível que o número resultante desta soma seja múltiplo de 5. $\blacksquare$ + +**d.** + +- Base da indução $(k = 1)$: + +$0^3 + 1^3 + 2^3 = 9 \equiv 0(\text{mod 9})$; + +- Hipótese de indução: Se + +$(k - 1)^3 + k^3 + (k + 1)^3 = k^3 - \cancel{3k^2} + 3k - \cancel 1 + k^3 + k^3 + \cancel{3k^2} + 3k - \cancel 1 \\ += 3k^3 + 6k \equiv 0(\text{mod 9})$para qualquer $k \in \N$, então também + +$3(k + 1)^3 + 6(k + 1) \equiv 0(\text{mod 9})$ + +- Passo de indução: + +$k^3 + (k + 1)^3 + (k + 2)^3 = k^3 + k^3 + 3k^2 + 3k + 1 + k^3 + 6k^2 + 12k + 8\\ = 3k^3 + 9k^2 + 9k + 3 + 6k + 6 = 3(k^3 + 3k^2 + 3k + 1) + 6k + 6\\ = 3(k + 1)^3 + 6(k + 1) \equiv \text{0(mod 9)}\,\blacksquare$ + +Logo, conclui-se que a soma de três cubos consecutivos de fato produz um número divisível por 9. + +### Exercício 9 + +- Base da indução $(n = 1)$: + +$(\cos \theta + i\sin \theta)^1 = \cos \theta + i\sin \theta$ + +- Hipótese de indução: se + +$(\cos \theta + i\sin \theta)^n = \cos (n\theta) + i \sin (n\theta)$ + + para todo $n \ge 1$, então também + +$(\cos \theta + i\sin \theta)^{n + 1} = \cos [(n + 1)\theta] + i \sin [(n + 1)\theta]$ + +- Passo de indução: + +$(\cos \theta + i\sin \theta)^{n + 1} = (\cos \theta + i\sin \theta)^n(\cos \theta + i\sin \theta)\\ = \underbrace{[\cos (n\theta) + i \sin (n\theta)]}_{\text{Por hipótese}}(\cos \theta + i\sin \theta)\\ = \cos(n\theta)\cos \theta - \sin(n\theta)\sin \theta + i[\cos(n\theta)\sin(\theta) + \sin(n\theta)\cos \theta]\\ = \underbrace{\cos [(n + 1)\theta] + i \sin [(n + 1)\theta]}_{\text{Por identidade trigonométrica}}$ + +**Fazendo uso da identidade de Euler** + +- Base da indução $(n = 1)$: + +$e^{1i\theta} = \cos \theta + i\sin \theta$ + +- Hipótese de indução: se + +$e^{ni\theta} = \cos (n\theta) + i \sin (n\theta)$ + +para todo $n \ge 1$, então também + +$e^{(n + 1)i\theta} = \cos [(n + 1)\theta] + i \sin [(n + 1)\theta]$ + +- Passo de indução: + +$e^{(n + 1)i\theta} = e^{ni\theta}e^{1i\theta} = (\cos \theta + i\sin \theta)^n(\cos \theta + i\sin \theta)\\ = \underbrace{[\cos (n\theta) + i \sin (n\theta)]}_{\text{Por hipótese}}(\cos \theta + i\sin \theta)\\ = \cos(n\theta)\cos \theta - \sin(n\theta)\sin \theta + i[\cos(n\theta)\sin(\theta) + \sin(n\theta)\cos \theta]\\ = \underbrace{\cos [(n + 1)\theta] + i \sin [(n + 1)\theta]}_{\text{Por identidade trigonométrica}}\,\blacksquare$ + +### Exercício 10 + +- Base da indução $(n = 1)$ + +$H_1(x) = e^{x^2}\left(- \dfrac d{dx}\right)^1e^{-x^2} = e^{x^2}\cdot 2xe^{-x^2} = 2x$ + +- Hipótese de indução: se + +$H_n(x) = e^{x^2}\left(-\dfrac d{dx}\right)^ne^{-x^2}$ + +para todo $n \ge 1$, então também + +$H_{n + 1}(x) = e^{x^2}\left(-\dfrac d{dx}\right)^{n + 1}e^{-x^2}$ + +- Passo de indução: + +$H_n(x) = e^{x^2}\left(-\dfrac d{dx}\right)^ne^{-x^2} \implies \left(\dfrac d{dx}\right)^ne^{-x^2} = (-1)^nH_n(x)e^{-x^2} \\ \left(\dfrac d{dx}\right)^{n + 1}e^{-x^2} = (-1)^n \underbrace{\left[\left(\dfrac d{dx} H_n (x)\right)e^{-x^2} + H_n(x)(-2x)e^{-x^2}\right]}_{\text{Aplicação da regra da cadeia}}\\ = (-1)^{n + 1}e^{-x^2}\left[\left(2x - \dfrac d{dx}\right)H_n(x)\right] = (-1)^{n + 1}e^{x^2}\left(\dfrac d{dx}\right)^{n + 1}e^{-x^2} = H_{n + 1}(x)\,\blacksquare$ + +Quanto a paridade do polinômio de Hermite, vemos que ele possui grau $n$, de tal sorte que, como demonstra o gráfico abaixo + + + +Este produz uma função ímpar quando $n$ é ímpar e uma função par quando não. + +### Exercício 11 + +- Base de indução $(n = 1)$: + +$\dfrac{[f(x)]'}{f(x)} = \dfrac{f'(x)}{f(x)}$ + +Para demonstrar a propriedade das derivadas subjacente a este cálculo, vale a pena explicitar o passo seguinte também. Considerando que + +$$ +f(x) = g(x) h(x) \implies f'(x) = g(x) h'(x) + g'(x) h(x) +$$ + +Então + +$\dfrac{[f_1(x)f_2(x)]'}{f_1(x)f_2(x)} = \dfrac{\cancel{f_1(x)}f'_2(x)}{\cancel{f_1(x)}f_2(x)} + \dfrac{\cancel{f_2(x)}f'_1(x)}{f_1(x)\cancel{f_2(x)}} = \dfrac{f'_1(x)}{f_1(x)} + \dfrac{f'_2(x)}{f_2(x)}$ + +- Hipótese de indução: se + +$\dfrac{[f_1(x)\dots f_n(x)]'}{f_1(x)\dots f_n(x)} = \dfrac{f_1'(x)}{f_1(x)} + \dots + \dfrac{f_n'(x)}{f_n(x)}$ + +então, para $n \ge 1$, $n \in \N$, + +$\dfrac{[f_1(x)\dots f_{n + 1}(x)]'}{f_1(x)\dots f_{n + 1}(x)} = \dfrac{f_1'(x)}{f_1(x)} + \dots + \dfrac{f_{n + 1}'(x)}{f_{n + 1}(x)}$ + +- Passo de indução: considerando $g(x) = f_1(x)\dots f_n$ + +$\dfrac{[g(x)f_{n + 1}]'}{g(x)f_{n + 1}} = \dfrac{\cancel{g(x)}f'_{n + 1}(x)}{\cancel{g(x)}f_{n + 1}(x)} + \dfrac{g'(x)\cancel{f_{n + 1}(x)}}{g(x)\cancel{f_{n + 1}(x)}} = \underbrace{\dfrac{f'_1(x)}{f_1(x)} + \dots + \dfrac{f'_n(x)}{f_n(x)}}_{\text{Por hipotése}} + \dfrac{f'_{n + 1}(x)}{f_{n + 1}(x)}\,\blacksquare$ + +### Exercício 12 + +Prosseguiremos nessa demonstração por absurdo. Assumiremos que $\sqrt 2 \in \mathbb Q$, portanto, $\sqrt 2 = \frac ab$, uma fração irredutível onde $a, b \in \Z$ e $b \not = 0$. + +**Lema:** Todo quadrado de um número inteiro não nulo tem 1 como resto da divisão inteira por três se não for divisível por 3. + +- Todo número inteiro quando dividido por três produz resto 0, 1 ou 2. Considerando que este não produza resto 0, temos: + + $\begin{cases} + (3k + 1)^2 = 9k^2 + 6k + 1 = 3\underbrace{(3k^2 + 2k)}_{k_2} + 1 \\ + (3k + 2)^2 = 9k^2 + 12k + 4 = 3\underbrace{(3k^2 + 4k + 1)}_{k_2} + 1 + \end{cases}, k \in \Z$ + +     + +Temos que $\left(\frac ab\right)^2 = 2$, 2 não é múltiplo de 3 e portanto + +$2 = 3k + 1 \implies 3k = 1 \implies k = \frac 13, k \not \in \Z$ + +O que é absurdo. **Teorema:** $\sqrt 2$ e irracional. $\blacksquare$ + +### Exercício 13 [^3] + +$(\sqrt 2^{1 + \sqrt 2})^{1 + \sqrt 2} = 2^\frac{(1 + \sqrt 2)^2}2 = \underbrace{2^\frac{3 + 2\sqrt 2}{2}}_{\text{irracional}} \blacksquare$ + +### Exercício 14 + +**a. Prova por indução finita comum** + +- Base de indução$(n = 2)$: 2 é primo: $2 = 2 \times 1$, então números primos existem. + +- Hipótese de indução: se um número $n > 1$, $n \in \N$ é primo ou múltiplo de primos, $n - 1$ também é. + +- Se $n + 1$ for primo, então não resta nada a provar. Senão, devemos concluir que existem pelo menos dois números inteiros $a$ e $b$, $1 < a \le b \le n$ tais que $ab = n$. Por vez estes também ou são primos ou múltiplos de primos, e assim por diante. Assim, percorremos todos os valores de $n$ à 1 e concluímos que a mesma condição se sustenta. $\blacksquare$ + +**b. Prova por indução forte** + +- Base de indução$(n = 2)$: 2 é primo: $2 = 2 \times 1$, então números primos existem. + +- Hipótese de indução: todos os números entre 1 e n ou são primos ou, senão, múltiplos de primos. + +- Passo de indução: Se $n$ for primo, então não resta nada a provar. Senão, devemos concluir que existem pelo menos dois números inteiros $a$ e $b$, $1< a \le b \le n$ tais que $ab = n$. Pela hipótese de indução, $a = p_1p_2\dots p_n$ e $b = q_1q_2\dots q_n$, sendo $p$ e $q$ números primos. Ora, então $n$ também é múltiplo de números primos: $n = p_1p_2\dots p_3q_1q_2\dots q_3$ $\blacksquare$ + +### Exercício 15 + + + +> Exemplo de desenho feito seguindo esse método para $n = 3$. + +**Propriedade 1:** Por não ser paralela a qualquer outra reta ou interceptá-las em um ponto comum, uma reta em posição geral intercepta demais retas no plano em n pontos distintos, sendo $n$ o número de demais retas. + +**Propriedade 2:** Por interceptar as $n$ retas, temos $n + 1$ subdivisões do plano em faces opostas, somadas ao número de faces anterior. Ou seja, + +$$ +F_{n + 1} = F_n + n + 1 +$$ + +1. Base de indução $(n = 0)$: + +$$ +\dfrac{n^2 + n + 2}2 = 1 +$$ + +De fato, um plano sem subdivisões tem uma única face. + +2. Hipótese de indução: se o número de faces de um plano dividido por retas de maneira a gerar o maior número de subdivisões é dado por + +$$ +F_n = \dfrac{n^2 + n + 2}2 +$$ + +Para qualquer número de retas $n$, então: + +$$ +F_{n + 1} = \dfrac{(n + 1)^2 (n + 1) + 2}2 +$$ + +Passo de indução: Retomando a fórmula dada pela propriedade 2: + +$F_{n + 1} = F_n + n + 1 = \dfrac{n^2 + n + 2}2 + n + 1 = \dfrac{n^2 + n + 2 + 2n + 2}2 = \\ = \dfrac{(n² + 2n + 1) + (n + 1) + 2}2 = \dfrac{(n + 1)^2 + (n + 1) + 2}2 \,\blacksquare$ + +### Exercício 16 + +- Base de indução: $0^p \equiv 0 (\text{mod p})$; + +- Hipótese de indução: se $n^p \equiv n (\text{mod }p)$ para qualquer $n \in \N$ então também $(n + 1)^p \equiv (n + 1)(\text{mod }p)$. + +- Passo de indução: para concluirmos, faremos uso do seguinte lema (o qual será por vez demonstrado ao final desta demonstração): + +$$ +(a + b)^p \equiv a^p + b^p (\text{mod }p), \forall a,b \in \N, p \in \mathbb P +$$ + +Ou seja, + +$(n + 1)^p \equiv n^p + 1^p(\text{mod }p) \equiv n^p + 1(\text{mod }p)$ + +Pela hipótese de indução $n^p \equiv n (\text{mod }p)$, então + +$$ +(n + 1)^p \equiv n + 1(\text{mod }p)\,\blacksquare +$$ + +**Demonstração do lema *Freshman's Dream* (o sonho do calouro):** + +Para $\forall a,b \in \N, p \in \mathbb P$, tem-se: + +$$ +(a + b)^p = a^p + \binom p1 x^{p - 1}y + \binom p2 x^{p - 2}y^2 + \dots + +\binom p{p - 1}xy^{p - 1} + y^p +$$ + +Isolando-se os coeficientes binomiais, tem-se que cada um deles pode ser escrito na forma: + +$$ +\binom pi = \frac{p!}{i!(p - i)!} = p \dfrac{(p - 1)!}{i! (p - i)!} \in \N +$$ + +Como tanto $i < p$ e $(p - i) < p$, e $p$ não é divisível senão por $p$, cada $\binom pi$ é um coeficiente múltiplo de $p$. Assim, o módulo de $(a + b)^p$ por $p$ é tal que: + +$$ +(a + b)^p \equiv a^p + b^p (\text{mod }p)\,\blacksquare +$$ + +### Exercício 17 + +Para analisar o tempo de processamento deste algoritmo, consideraremos a linha 1 que se repete $(n - 1)$ vezes. A operação executada nessa linha, porém, não é atômica: ela toma tempo proporcional ao tamanho da entrada que varia em cada iteração. Na primeira iteração a comparação na linha 4 é feita $(n - 1)$ vezes, na segunda $(n − 2)$, e assim por diante. Assim, o tempo de processamento em função do tamanho $n$ da entrada, então, é: + +$ \displaystyle \sum^{n - 1}_{i = 1}(n - i) += \sum^{n - 1}_{i = 1} n - \sum^{n - 1}_{i = 1} i += n(n - 1) - \frac{(n - 1)[(n - 1) + 1]}2 \\ += (n - 1)\left(n - \frac n2\right) = \frac{n (n - 1)}2 += \dfrac{n^2 - n}2 \in \Theta(n^2)\,\blacksquare +$ + +[^1]: nUSP: 12543033; Turma 04. + +[^2]: Resposta retirada diretamente das notas de aula do dia 03/09/2021 + +[^3]: $(\sqrt 2^{\sqrt 2})^{\sqrt 2} = (2^{\frac 12 \cdot 2^{\frac 12}})^{2^{\frac 12}} = 2^{\cancel{\frac 12 \cdot 2^{\frac12} \cdot 2^{\frac 12}}} = 2$ \ No newline at end of file diff --git a/Matemática Discreta I/Lista 3/Imagens/0666956f5b22720e81a8b19e0123d0bbb181e628.png b/Matemática Discreta I/Lista 3/Imagens/0666956f5b22720e81a8b19e0123d0bbb181e628.png new file mode 100644 index 0000000000000000000000000000000000000000..a569e01a0bda08e0ba1fc9d621dd43130c92ef65 GIT binary patch literal 407803 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