From 5d794e71372991c3add4e47ef4ec17921e5fbfdf Mon Sep 17 00:00:00 2001 From: saundersp Date: Fri, 20 Dec 2024 21:34:20 +0100 Subject: [PATCH] contents/computer_science.tex : Better definitions and description && Added AOC implies LEM proof --- contents/set_theory.tex | 102 ++++++++++++++++++++++++++++++++++++---- 1 file changed, 92 insertions(+), 10 deletions(-) diff --git a/contents/set_theory.tex b/contents/set_theory.tex index 52884b5..48872b9 100644 --- a/contents/set_theory.tex +++ b/contents/set_theory.tex @@ -25,11 +25,34 @@ Il existe un ensemble vide notée $\emptyset$. \langsubsection{Paire}{Pairing} %TODO Complete subsection +Source: \citeannexes{wikipedia_ordered_pair} + +\langsubsubsection{Définition de Wiener}{Wiener's definition} + +$(a,b) := \{\{\{a\}, \emptyset\}, \{b\}\}$ + +\langsubsubsection{Définition de Hausdorff}{Hausdorff's definition} + +$(a,b) := \{\{a, 1\}, \{b,2\}\}$ where $a \ne 1 \land b \ne 2$ + +\langsubsubsection{Définition de Kuratowski}{Kuratowski's definition} + +\begin{definition_sq} \label{definition:ordered_pair} +$(a,b)_K := \{\{a\}, \{a,b\}\}$ +\end{definition_sq} + \langsubsection{Réunion}{Union} -%TODO Complete section Unite all elements of two given sets into one. +\begin{definition_sq} \label{definition:set_union} +$A \union B := \{x | (x \in A \lor x \in B)\}$ +\end{definition_sq} + +Pour des ensembles finis : $\forall E,F \in \Cat(\Set), \card{E \union F} = \card{E} + \card{F} - \card{E \intersection F}$ + +Example : + $n,m \in \N$ $A := \{a_0, \cdots, a_n\}$ @@ -52,28 +75,81 @@ For a set $S$ such that $\card{S} = n \implies \card{\mathbf{P}(S)} = 2^n$ \langsubsection{Choix}{Choice} %TODO Complete subsection +\begin{definition_sq} \label{definition:set_axiom_of_choice} +For any set $X$ of nonempty sets, there exists a choice function $f$ that is defined on $X$ and maps each set of $X$ to an element of that set i.e. + +$$\forall X [\emptyset \notin X \implies \exists \function{f}{X}{\Union_{A \in X} A \quad \forall A \in X(f(A) \in A)}]$$ +\end{definition_sq} + +\begin{theorem_sq} \label{theorem:ac_implies_lem} +The axiom of choice implies the law of excluding middle. +\end{theorem_sq} + +\begin{proof} + +Assume that $0 \ne 1$ (or any two elements that are not equal), Let $\Omega := \{0, 1\}$, $p \in \mathbf{Prop}$ + +$A := \{ x \in \Omega | x = 0 \lor p \}$ + +$B := \{ y \in \Omega | y = 1 \lor p \}$ + +$\implies 0 \in A \land 1 \in B$ + +$X := \{ A, B \}$, by definition $\Union X = \Omega$ + +By the axiom of choice $\implies \exists \function{f}{X}{\Omega}$ + +Using this function there are 4 cases: +\begin{enumerate}[(1)] + \item $f(A) = f(B) = 0 \implies 0 \in B$ but $((0 = 1) \lor p \implies \top) \implies p$ + \item $f(A) = f(B) = 1$ Same reasoning as (1) $\implies p$ % TODO Replace with local labeling and reference + \item $f(A) \neq f(B) = 0 \implies A \neq B$ but $p \implies A = B = \Omega$ (contrapositive of (1) and (2)) $\implies \lnot p$ + \item $f(A) \neq f(B) = 1$ Same reasoning as (3) $\implies \lnot p$ +\end{enumerate} + +So by proof by cases $(p \lor \lnot p)$ which is the law of excluded middle \ref{definition:law_excluding_middle}. +\end{proof} + \section{Intersection} Unite all common elements of two given sets into one. -$n,m,i \in \N$ +\begin{definition_sq} \label{definition:set_intersection} +$A \intersection B := \{x | (x \in A \land x \in B)\}$ +\end{definition_sq} -$A = \{a_0, \cdots, a_n, c_0, \cdots, c_n\}$ +Pour des ensembles finis : $\forall E,F \in \Cat(\Set), \card{E \intersection F} = \card{E} - \card{F} + \card{E \union F}$ -$B = \{b_0, \cdots, b_m, c_0, \cdots, c_n\}$ +Example : -$A \cap B = \{c_0, \cdots, c_n\}$ +$n,m \in \N$ + +$A := \{a_0, \cdots, a_n, c_0, \cdots, c_n\}$ + +$B := \{b_0, \cdots, b_m, c_0, \cdots, c_n\}$ + +$A \intersection B = \{c_0, \cdots, c_n\}$ \langsection{Différence des sets}{Set difference} %TODO Complete section +Exclude elements of a set from a set + +\begin{definition_sq} \label{definition:set_difference} +$A \setminus B := \{x | (x \in A \land x \notin B)\}$ +\end{definition_sq} + +Pour des ensembles finis : $\forall E,F \in \Cat(\Set), \card{E \setminus F} = \card{E} - \card{E \intersection F}$ + \langsection{Fonction}{Function} Source: \citeannexes{wikipedia_function_mathematics} +\begin{definition_sq} \label{definition:set_function} Une fonction $f$ est un tuple d'un domaine \citeannexes{wikipedia_domain_function} $A$ et un codomaine \citeannexes{wikipedia_codomain} $B$. +\end{definition_sq} -If the domain is the same as the codomain then the function is an endormorphsim \ref{definition:endomorphism} applied on set theory \ref{set_theory}. +If the domain is the same as the codomain then the function is an endormorphsim \ref{definition:endomorphism} applied the category \ref{definition:category} of sets \ref{set_theory}. \subsection{Notation} @@ -81,22 +157,28 @@ $\functiondef{A}{B}$ $\function{f}{x}{f(x)}$ -\langsubsection{Injectivité}{Injectivity} \label{definition:injective} +\langsubsection{Injectivité}{Injectivity} Source: \citeannexes{wikipedia_injective_function} +\begin{definition_sq} \label{definition:injective} Une fonction $f$ de $E$ dans $F$ est dite \textbf{injective} si, et seulement si, $\forall (a,b) \in E, f(a) = f(b) \Rightarrow a = b$. +\end{definition_sq} -\langsubsection{Surjectivité}{Surjectivity} \label{definition:surjective} +\langsubsection{Surjectivité}{Surjectivity} Source: \citeannexes{wikipedia_surjective_function} +\begin{definition_sq} \label{definition:surjective} Une fonction $f$ de $E$ dans $F$ est dite \textbf{surjective} si, et seulement si, $\forall y \in F, \exists x \in E : y = f(x)$. +\end{definition_sq} \langsubsection{Bijectivité}{Bijectivity} -Source: \citeannexes{wikipedia_bijection} \label{definition:bijection} +Source: \citeannexes{wikipedia_bijection} +\begin{definition_sq} \label{definition:bijection} Une fonction $f$ de $E$ dans $F$ est dite \textbf{bijective} si, et seulement si, elle est à la fois injective \ref{definition:injective} et surjective \ref{definition:surjective} ou $\forall y \in F, \exists! x \in E : y = f(x)$. +\end{definition_sq} -Every bijection is an isomorphism \ref{definition:isomorphism} applied on set theory \ref{set_theory}. +Every bijection is an isomorphism \ref{definition:isomorphism} applied on the category \ref{definition:category} of sets \ref{set_theory}.