211 lines
6.6 KiB
TeX
211 lines
6.6 KiB
TeX
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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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