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\documentclass{article}
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\usepackage{mathtools}
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\usepackage{amssymb}
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\title{Algèbre 1 - TD 1 \\ NON CORRIGÉ}
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\author{Timéo Pochin}
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\begin{document}
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\maketitle
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\section*{Exercice 1}
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\subsection*{1)}
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\begin{align}
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& \neg ( P \land \neg Q ) \nonumber \\
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\iff\quad & \neg P \lor Q \nonumber
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\end{align}
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\subsection*{2)}
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\begin{align}
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& \neg\big(P\land (Q\land R)\big) \nonumber \\
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\iff\quad & \neg ( P \land Q \land R ) \nonumber \\
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\iff\quad & \neg P \lor \neg Q \lor \neg R \nonumber
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\end{align}
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\subsection*{3)}
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\begin{align}
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& \neg\big(P\lor (Q\land R)\big) \nonumber \\
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\iff\quad & \neg P \land \neg (Q \land R) \nonumber \\
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\iff\quad & \neg P \land (\neg Q \lor \neg R) \nonumber
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\end{align}
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\subsection*{4)}
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\begin{align}
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& \neg\big((P\land Q)\Rightarrow(R\Rightarrow S)\big) \nonumber \\
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\iff\quad & P \land Q \land \neg (R \Rightarrow S) \nonumber \\
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\iff\quad & P \land Q \land R \land \neg S \nonumber
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\end{align}
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\section*{Exercice 2}
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\subsection*{}
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\begin{tabular}{ |c|c|c||c|c|c| }
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\hline
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$P$ & $Q$ & $R$ & $\neg P$ & $\neg Q$ & $\neg P \Leftrightarrow \neg Q$ \\ \hline
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\hline
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F & F & F & V & V & V \\ \hline
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F & F & V & V & V & V \\ \hline
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V & V & F & F & F & V \\ \hline
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V & V & V & F & F & V \\ \hline
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\end{tabular}
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\subsection*{}
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\begin{tabular}{ |c|c|c||c|c|c| }
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\hline
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$P$ & $Q$ & $R$ & $P\Rightarrow R$ & $Q\Rightarrow R$ & $(P\Rightarrow R)\Leftrightarrow(Q\Rightarrow R)$ \\ \hline
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\hline
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F & F & F & V & V & V \\ \hline
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F & F & V & V & V & V \\ \hline
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V & V & F & F & F & V \\ \hline
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V & V & V & V & V & V \\ \hline
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\end{tabular}
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\subsection*{}
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\begin{tabular}{ |c|c|c||c|c|c| }
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\hline
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$P$ & $Q$ & $R$ & $R\Rightarrow P$ & $R\Rightarrow Q$ & $(R\Rightarrow P)\Leftrightarrow(R\Rightarrow Q)$ \\ \hline
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\hline
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F & F & F & V & V & V \\ \hline
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F & F & V & F & F & V \\ \hline
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V & V & F & V & V & V \\ \hline
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V & V & V & V & V & V \\ \hline
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\end{tabular}
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\subsection*{}
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\begin{tabular}{ |c|c|c||c|c|c| }
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\hline
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$P$ & $Q$ & $R$ & $P\land R$ & $Q\land R$ & $(P\land R)\Leftrightarrow(Q\land R)$ \\ \hline
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\hline
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F & F & F & F & F & V \\ \hline
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F & F & V & F & F & V \\ \hline
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V & V & F & F & F & V \\ \hline
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V & V & V & V & V & V \\ \hline
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\end{tabular}
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\subsection*{}
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\begin{tabular}{ |c|c|c||c|c|c| }
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\hline
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$P$ & $Q$ & $R$ & $P\lor R$ & $Q\lor R$ & $(P\lor R)\Leftrightarrow(Q\lor R)$ \\ \hline
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\hline
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F & F & F & F & F & V \\ \hline
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F & F & V & V & V & V \\ \hline
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V & V & F & V & V & V \\ \hline
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V & V & V & V & V & V \\ \hline
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\end{tabular}
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\section*{Exercice 3}
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\subsection*{1)}
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\begin{tabular}{ |c|c||c|c|c|c|c|c| }
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\hline
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$P$ & $Q$ & $P\lor Q$ & $\neg(P\lor Q)$ & $\neg P$ & $\neg Q$ & $\neg P\land\neg Q$ & $\neg(P\lor Q)\Leftrightarrow\neg P\land\neg Q$ \\ \hline
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\hline
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F & F & F & V & V & V & V & V \\ \hline
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F & V & V & F & V & F & F & V \\ \hline
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V & F & V & F & F & V & F & V \\ \hline
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V & V & V & F & F & F & F & V \\ \hline
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\end{tabular}
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\subsection*{2)}
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\begin{tabular}{ |c|c|c||c|c|c|c|c| }
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\hline
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$P$ & $Q$ & $R$ & $P\Rightarrow Q$ & $Q\Rightarrow R$ & $(P\Rightarrow Q)\land(Q\Rightarrow R)$ & $P\Rightarrow R$ & $\big((P\Rightarrow Q)\land(Q\Rightarrow R)\big)\Rightarrow(P\Rightarrow R)$ \\ \hline
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\hline
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%P Q R
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F & F & F & V & V & V & V & V \\ \hline
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F & F & V & V & V & V & V & V \\ \hline
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F & V & F & V & F & F & V & V \\ \hline
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F & V & V & V & V & V & V & V \\ \hline
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V & F & F & F & V & F & F & V \\ \hline
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V & F & V & F & V & F & V & V \\ \hline
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V & V & F & V & F & F & F & V \\ \hline
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V & V & V & V & V & V & V & V \\ \hline
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\end{tabular}
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\section*{Exercice 4}
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\subsection*{1)}
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\begin{align}
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& (P\land Q)\Rightarrow P \nonumber \\
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\iff\quad & \neg(P\land Q)\lor P \nonumber \\
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\iff\quad & \neg P\lor\neg Q\lor P \nonumber \\
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\iff\quad & \text{Vrai}\lor\neg Q \nonumber \\
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\iff\quad & \text{Vrai} \nonumber
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\end{align}
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\subsection*{2)}
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\begin{align}
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& \bigg(\big(P\Rightarrow(Q\Rightarrow R)\big)\Leftrightarrow\big((P\land Q)\Rightarrow R\big)\bigg) \nonumber \\
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\iff\quad & \bigg(\big(\neg P\lor(Q\Rightarrow R)\Leftrightarrow\neg(P\land Q)\lor R\big)\bigg) \nonumber \\
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\iff\quad & (\neg P\lor\neg Q\lor R\Leftrightarrow\neg P\lor\neg Q\lor R) \nonumber \\
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\iff\quad & \text{Vrai} \nonumber
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\end{align}
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\subsection*{3)}
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\begin{align}
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& \bigg(\big((P\land Q)\Rightarrow R\big)\Leftrightarrow\big(\neg P\lor(Q\Rightarrow R)\big)\bigg) \nonumber \\
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\iff\quad & \bigg(\big(\neg(P\land Q)\lor R\big)\Leftrightarrow(\neg P\lor\neg Q\lor R)\bigg) \nonumber \\
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\iff\quad & \big((\neg P\lor\neg Q\lor R)\Leftrightarrow(\neg P\lor\neg Q\lor R)\big) \nonumber \\
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\iff\quad & \text{Vrai} \nonumber
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\end{align}
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\section*{Exercice 5}
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\subsection*{1)}
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Faux
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\subsection*{2)}
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Vrai
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\subsection*{3)}
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Faux
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\section*{Exercice 6}
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\subsection*{1)}
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Il existe un triangle rectangle qui ne possède pas un angle droit.
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\subsection*{2)}
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Il existe une maison où au moins un enfant n’aime pas ses deux parents.
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\subsection*{3)}
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Il existe un entier $x$ tel que, pour tout entier $y$, pour tout entier $z$, la relation $z\geq x+1$ implique la relation $z\geq y$.
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\subsection*{4)}
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\[
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\exists\epsilon>0,\quad \forall\alpha>0,\quad |5x-7|\geq\epsilon\Rightarrow|x-\frac{7}{5}|\geq\alpha
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\]
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\section*{Exercice 7}
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\subsection*{1)}
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\[
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\forall n\in\mathbb{R}^+,\quad \exists a\in\{x\in\mathbb{R}\mid-n<x<n\},\quad a\in A
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\]
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\subsection*{2)}
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\[
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\{0\}
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\]
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\subsection*{3)}
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\[
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\{1\}
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\]
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\section*{Exercice 8}
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\subsection*{1)}
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Pour tout entier naturel $n$, il existe un entier naturel $p$ tel que, $n$ est plus petit ou égal à $p$.
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\\\\
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Il existe un entier naturel $n$ tel que, pour tout entier naturel $p$, $n$ est plus grand que $p$.
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\subsection*{2)}
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Pour tout entier naturel $p$, pour tout entier négatif $n$, $p$ est plus grand ou égal à $n$.
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\\\\
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Il existe un entier naturel $p$ et un entier négatif $n$ tel que, $p$ est plus petit que $n$.
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\end{document}
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@ -12,24 +12,83 @@
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\section*{Exercice 1}
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\subsection*{a)}
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\begin{align}
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& 5x+2\geq-3 \nonumber \\
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\iff\quad & 5x\geq-5 \nonumber \\
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\iff\quad & x\geq-1 \nonumber \\
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\iff\quad & x=\left[-1,+\infty\right[ \nonumber
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\end{align}
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\subsection*{b)}
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\begin{align}
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& 2x-1<4x+3\leq-x+6 \nonumber \\
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\iff\quad & (2x-1<4x+3)\land(4x+3\leq-x+6) \nonumber \\
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\iff\quad & (-4<2x)\land(5x\leq 3) \nonumber \\
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\iff\quad & (-2<x)\land(x\leq\frac{3}{5}) \nonumber \\
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\iff\quad & x=\left]-2,\frac{3}{5}\right] \nonumber
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\end{align}
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\subsection*{c)}
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\begin{align}
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& |x-1|<4 \nonumber \\
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\iff\quad & (x-1<4)\land(x-1>-4) \nonumber \\
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\iff\quad & (x<5)\land(x>-3) \nonumber \\
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\iff\quad & x=\left]-3,5\right[ \nonumber
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\end{align}
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\subsection*{d)}
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\begin{align}
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& |x-2|\geq 3 \nonumber \\
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\iff\quad & (x-2\geq 3)\lor(x-2\leq-3) \nonumber \\
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\iff\quad & (x\geq 5)\lor(x\leq-1) \nonumber \\
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\iff\quad & x=\left]-\infty,-1\right]\ \cup\ \left[5,+\infty\right[ \nonumber
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\end{align}
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\subsection*{e)}
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\begin{align}
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& |x-2|\leq|x| \nonumber \\
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\iff\quad & (x-2)^2\leq x^2 \nonumber \\
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\iff\quad & x^2-4x+4\leq x^2 \nonumber \\
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\iff\quad & 4\leq 4x \nonumber \\
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\iff\quad & 1\leq x \nonumber \\
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\iff\quad & x=\left[1,+\infty\right[ \nonumber
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\end{align}
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\subsection*{f)}
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\subsection*{g)}
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\begin{align}
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& \sqrt{x+1}<2 \nonumber \\
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\iff\quad & (x+1<4)\land(x+1\geq0) \nonumber \\
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\iff\quad & (x<3)\land(x\geq-1) \nonumber \\
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\iff\quad & x=\left[-1,3\right[ \nonumber
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\end{align}
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\subsection*{h)}
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\begin{align}
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& x^2+1\leq 3 \nonumber \\
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\iff\quad & x^2\leq 2 \nonumber \\
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\iff\quad & (x\leq\sqrt{2})\land(x\geq-\sqrt{2}) \nonumber \\
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\iff\quad & x=\left[-\sqrt{2},\sqrt{2}\right] \nonumber
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\end{align}
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\subsection*{i)}
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\begin{align}
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& x^2+3x<4 \nonumber \\
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\iff\quad & x^2+3x-4<0 \nonumber \\
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\iff\quad & (x-1)(x+4)<0 \nonumber \\
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\iff\quad & (x<1)\land(x>-4) \nonumber \\
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\iff\quad & x=\left]-4,1\right[ \nonumber
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\end{align}
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\subsection*{j)}
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\begin{align}
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& x^3-3x^2+2x\geq 0 \nonumber \\
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\iff\quad & x(x^2-3x+2)\geq 0 \nonumber \\
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\iff\quad & x(x-1)(x-2)\geq 0 \nonumber \\
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\iff\quad & \big((x\geq 0)\land(x\leq 1)\big)\lor(x\geq 2) \nonumber \\
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\iff\quad & x=\left[0,1\right]\cup\left[2,+\infty\right[ \nonumber
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\end{align}
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\subsection*{k)}
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@ -0,0 +1,268 @@
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\documentclass{article}
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\usepackage{mathtools}
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\usepackage{amssymb}
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\title{Mécanique du point - TD 1 \\ NON CORRIGÉ}
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\author{Timéo Pochin}
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\begin{document}
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\maketitle
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\section*{Exercice 1}
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\subsection*{a)}
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\begin{align}
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\vec{u}\land\vec{v}
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\nonumber
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&=
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\begin{pmatrix}
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1 \\
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2 \\
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3
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\end{pmatrix}
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\land
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\begin{pmatrix}
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4 \\
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5 \\
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6
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\end{pmatrix}
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\\ \nonumber
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&=
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\begin{pmatrix}
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2\cdot 6-5\cdot 3 \\
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4\cdot 3-1\cdot 6 \\
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1\cdot 5-4\cdot 2
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\end{pmatrix}
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\\ \nonumber
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&=
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\begin{pmatrix}
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-3 \\
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6 \\
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-3
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\end{pmatrix}
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\\ \nonumber
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&=-3\vec{e_x}+6\vec{e_y}-3\vec{e_z}
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\end{align}
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\subsection*{b)}
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\begin{align}
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\vec{v}\land\vec{u}
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\nonumber
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&=
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\begin{pmatrix}
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4 \\
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5 \\
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6
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\end{pmatrix}
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\land
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\begin{pmatrix}
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1 \\
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2 \\
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3
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\end{pmatrix}
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\\ \nonumber
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&=
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\begin{pmatrix}
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5\cdot 3-2\cdot 6 \\
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1\cdot 6-4\cdot 3 \\
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4\cdot 2-1\cdot 5
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\end{pmatrix}
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\\ \nonumber
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&=
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\begin{pmatrix}
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3 \\
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-6 \\
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3
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\end{pmatrix}
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\\ \nonumber
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&=3\vec{e_x}-6\vec{e_y}+3\vec{e_z}
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\end{align}
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\subsection*{c)}
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\begin{align}
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\vec{u}\cdot\vec{v}
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\nonumber
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&=
|
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\begin{pmatrix}
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1 \\
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2 \\
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3
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\end{pmatrix}
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\cdot
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\begin{pmatrix}
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4 \\
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5 \\
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6
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\end{pmatrix}
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\\ \nonumber
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&=1\cdot 4+2\cdot 5+3\cdot 6\\ \nonumber
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&=32
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\end{align}
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\subsection*{d)}
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\begin{align}
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\vec{u}\cdot(\vec{u}\land\vec{v})
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\nonumber
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&=
|
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\begin{pmatrix}
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1 \\
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2 \\
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3
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\end{pmatrix}
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\cdot
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\begin{pmatrix}
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-3 \\
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6 \\
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-3
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\end{pmatrix}
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\\ \nonumber
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&=1\cdot-3+2\cdot 6+3\cdot -3\\ \nonumber
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&=0
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\end{align}
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\subsection*{e)}
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\begin{align}
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\|\vec{u}\|
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\nonumber
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&=\sqrt{1^2+2^2+3^2}
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\\ \nonumber
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&=\sqrt{14}
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\end{align}
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\subsection*{f)}
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\begin{align}
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\|\vec{u}+\vec{v}\|
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\nonumber
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&=
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\left\|
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\begin{pmatrix}
|
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1 \\
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2 \\
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3
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\end{pmatrix}
|
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+
|
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\begin{pmatrix}
|
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4 \\
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5 \\
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6
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\end{pmatrix}
|
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\right\|
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\\ \nonumber
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&=
|
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\left\|
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\begin{pmatrix}
|
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5 \\
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7 \\
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9
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\end{pmatrix}
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\right\|
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\\ \nonumber
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&=\sqrt{5^2+7^2+9^2}
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\\ \nonumber
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||||
&=\sqrt{155}
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\end{align}
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\subsection*{g)}
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\begin{align}
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(\vec{u}\land\vec{v})\land\vec{w}
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\nonumber
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&=
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||||
\begin{pmatrix}
|
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-3 \\
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6 \\
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-3
|
||||
\end{pmatrix}
|
||||
\land
|
||||
\begin{pmatrix}
|
||||
7 \\
|
||||
8 \\
|
||||
9
|
||||
\end{pmatrix}
|
||||
\\ \nonumber
|
||||
&=
|
||||
\begin{pmatrix}
|
||||
6\cdot 9-8\cdot-3 \\
|
||||
7\cdot-3-9\cdot-3 \\
|
||||
-3\cdot 8-7\cdot 6
|
||||
\end{pmatrix}
|
||||
\\ \nonumber
|
||||
&=
|
||||
\begin{pmatrix}
|
||||
78 \\
|
||||
6 \\
|
||||
-66
|
||||
\end{pmatrix}
|
||||
\\ \nonumber
|
||||
&=78\vec{e_x}+6\vec{e_y}-66\vec{e_z}
|
||||
\end{align}
|
||||
|
||||
\subsection*{h)}
|
||||
\begin{align}
|
||||
\vec{u}\land(\vec{v}\land\vec{w})
|
||||
\nonumber
|
||||
&=
|
||||
\begin{pmatrix}
|
||||
1 \\
|
||||
2 \\
|
||||
3
|
||||
\end{pmatrix}
|
||||
\land
|
||||
\left(
|
||||
\begin{pmatrix}
|
||||
4 \\
|
||||
5 \\
|
||||
6
|
||||
\end{pmatrix}
|
||||
\land
|
||||
\begin{pmatrix}
|
||||
7 \\
|
||||
8 \\
|
||||
9
|
||||
\end{pmatrix}
|
||||
\right)
|
||||
\\ \nonumber
|
||||
&=
|
||||
\begin{pmatrix}
|
||||
1 \\
|
||||
2 \\
|
||||
3
|
||||
\end{pmatrix}
|
||||
\land
|
||||
\begin{pmatrix}
|
||||
5\cdot 9-8\cdot 6 \\
|
||||
7\cdot 6-4\cdot 9 \\
|
||||
4\cdot 8-7\cdot 5
|
||||
\end{pmatrix}
|
||||
\\ \nonumber
|
||||
&=
|
||||
\begin{pmatrix}
|
||||
1 \\
|
||||
2 \\
|
||||
3
|
||||
\end{pmatrix}
|
||||
\land
|
||||
\begin{pmatrix}
|
||||
-3 \\
|
||||
6 \\
|
||||
-3
|
||||
\end{pmatrix}
|
||||
\\ \nonumber
|
||||
&=
|
||||
\begin{pmatrix}
|
||||
2\cdot-3-6\cdot 3 \\
|
||||
-3\cdot 3-1\cdot-3 \\
|
||||
1\cdot 6--3\cdot 3
|
||||
\end{pmatrix}
|
||||
\\ \nonumber
|
||||
&=
|
||||
\begin{pmatrix}
|
||||
-24 \\
|
||||
-6 \\
|
||||
12
|
||||
\end{pmatrix}
|
||||
\\ \nonumber
|
||||
&= -24\vec{e_x}-6\vec{e_y}+12\vec{e_z}
|
||||
\end{align}
|
||||
|
||||
\end{document}
|
||||
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Reference in New Issue