contents/logic.tex : Updated symbols and subsubsection
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\langchapter{Logique}{Logic}
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%TODO Complete chapter
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La logique consiste en des opérations effectuées uniquement sur des variables (notées $P,Q,R$) n'ayant pour valeur soit Vrai (noté \true), soit Faux (noté \false).
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%Logic consists of operations done on sole values : True $T$ and False $F$.
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\lang{La logique classique consiste en des opérations effectuées uniquement sur des propositions (typiquement notées $p$ ou $q$) n'ayant pour valeur soit Vrai (noté \true), soit Faux (noté \false).}%
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{Classical logic consists of operations done on sole values : True $T$ and False $F$.}
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\langsection{Principle de tiers exclu}{Excluding middle}
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\langsection{Principle de tiers exclu}{Excluding middle} \label{definition:law_excluding_middle}
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$\true \equivalence \lnot \false$
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$\false \equivalence \lnot \true$
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$\lnot\lnot p \implies p$
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$p \lor \lnot p$
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\langsection{Relation Binaires}{Binary relations}
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%TODO Complete section
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\langsubsection{Réflexion}{Reflexivity} \label{definition:reflexivity}
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% TODO Complete subsection
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Une relation $\Rel$ sur $E$ est dite \textbf{réflexive} si et seulement si $\forall a \in E, a \Rel a$.
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Une relation $\Rel$ sur $E$ est dite \textbf{réflexive} si et seulement si $\forall a \in E$, $a \Rel a$.
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\langsubsection{Transitivité}{Transitivity} \label{definition:transitivity}
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% TODO Complete subsection
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@ -31,21 +35,21 @@ Une relation $\Rel$ sur $E$ est dite \textbf{associative} si et seulement si $\f
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\langsubsection{Commutativité}{Commutativity} \label{definition:commutativity}
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% TODO Complete subsection
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Une relation $\Rel$ sur $E$ est dite \textbf{commutative} si et seulement si $\forall (a,b) \in E, a \Rel b = b \Rel a$.
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Une relation $\Rel$ sur $E$ est dite \textbf{commutative} si et seulement si $\forall (a,b) \in E$, $a \Rel b = b \Rel a$.
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\langsection{Opérateurs}{Operators}
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%TODO Complete section
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\langsubsection{NON}{NOT}
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\langsubsection{NON $(\lnot)$}{NOT $(\lnot)$}
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% TODO Complete subsection
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$P \equivalence \lnot \lnot P$
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$p \equivalence \lnot \lnot p$
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\langsubsubsection{Table de vérité}{Truth table}
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\begin{tabular}{|c|c|}
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\hline
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P & $\lnot P$ \\
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$p$ & $\lnot p$ \\
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\hline
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\false & \true \\
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\hline
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@ -53,14 +57,16 @@ $P \equivalence \lnot \lnot P$
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\hline
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\end{tabular}
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\langsubsection{ET}{AND}
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\langsubsection{ET $(\land)$}{AND $(\land)$}
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%TODO Complete subsection
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$P \land Q \equivalence \lnot P \lor \lnot Q$
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$p \land q \equivalence \lnot p \lor \lnot q$
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\langsubsubsection{Table de vérité}{Truth table}
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\begin{tabular}{|c|c||c|}
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\hline
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P & Q & P $\land$ Q \\
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$p$ & $q$ & $p \land q$ \\
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\hline
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\false & \false & \false \\
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\hline
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@ -72,16 +78,16 @@ $P \land Q \equivalence \lnot P \lor \lnot Q$
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\hline
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\end{tabular}
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\langsubsection{OU}{OR}
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\langsubsection{OU $(\lor)$}{OR $(\lor)$}
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% TODO Complete subsection
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$P \lor Q \equivalence \lnot P \land \lnot Q$
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$p \lor q \equivalence \lnot p \land \lnot q$
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\medskip
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\langsubsubsection{Table de vérité}{Truth table}
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\begin{tabular}{|c|c||c|}
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\hline
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P & Q & P $\lor$ Q \\
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$p$ & $q$ & $p \lor q$ \\
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\hline
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\false & \false & \false \\
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\hline
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@ -93,12 +99,14 @@ $P \lor Q \equivalence \lnot P \land \lnot Q$
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\hline
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\end{tabular}
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\subsection{Implication}
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\subsection{Implication $(\implies)$}
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%TODO Complete subsection
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\langsubsubsection{Table de vérité}{Truth table}
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\begin{tabular}{|c|c||c|}
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\hline
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P & Q & P $\Rightarrow$ Q \\
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$p$ & $q$ & $p \implies q$ \\
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\hline
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\false & \false & \true \\
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\hline
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@ -110,15 +118,32 @@ $P \lor Q \equivalence \lnot P \land \lnot Q$
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\hline
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\end{tabular}
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\lang{Contraposée}{Contraposition } : \
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$\lnot Q \implies \lnot P$
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\lang{Contraposée}{Contraposition} : $\lnot q \implies \lnot p$
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\langsubsection{Équivalence}{Equivalence}
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% TODO Complete subsection
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\langsubsubsection{Table de vérité}{Truth table}
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\begin{tabular}{|c|c||c|}
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\hline
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$P$ & $Q$ & $P \equivalence Q$ \\
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$p$ & $q$ & $p \implies q$ \\
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\hline
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\false & \false & \true \\
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\hline
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\true & \false & \false \\
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\hline
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\false & \true & \true \\
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\hline
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\true & \true & \true \\
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\hline
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\end{tabular}
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\langsubsection{Équivalence $(\equivalence)$}{Equivalence $(\equivalence)$}
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% TODO Complete subsection
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\langsubsubsection{Table de vérité}{Truth table}
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\begin{tabular}{|c|c||c|}
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\hline
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$p$ & $q$ & $p \equivalence q$ \\
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\hline
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\false & \false & \true \\
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\hline
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@ -130,14 +155,16 @@ $\lnot Q \implies \lnot P$
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\hline
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\end{tabular}
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\langsubsection{OU exclusif / XOR}{Exclusive OR / XOR}
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\langsubsection{OU exclusif / XOR $(\oplus)$}{Exclusive OR / XOR $(\oplus)$}
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%TODO Complete subsection
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$P \oplus Q \equivalence (P \lor Q) \land \lnot (P \land Q)$
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$p \oplus q \equivalence (p \lor q) \land \lnot (p \land q)$
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\langsubsubsection{Table de vérité}{Truth table}
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\begin{tabular}{|c|c||c|}
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\hline
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P & Q & $P \oplus Q$ \\
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$p$ & $q$ & $p \oplus q$ \\
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\hline
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\false & \false & \false \\
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\hline
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