185 lines
5.9 KiB
TeX
185 lines
5.9 KiB
TeX
\langchapter{Théorie des ensembles}{Set theory} \label{set_theory}
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%TODO Complete chapter
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Source: \citeannexes{wikipedia_set_theory}
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Un ensemble est une construction mathématique qui réuni plusieurs objets en une même instance.
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%A set is a mathematical construct to assemble multiple objects in a single instance.
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$S = \{a, b, c\}$
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\langsection{Axiomes}{Axioms}
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%TODO Complete section
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\langsubsection{Extensionnalité}{Extensionality}
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$\forall A\forall B(\forall X(X \in A \equivalence X \in B) \implies A = B)$
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\langsubsection{Spécification}{Specification}
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%TODO Complete subsection
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\langsubsection{Ensemble vide}{Empty set}
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Il existe un ensemble vide noté $\emptyset$.
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\langsubsection{Paire}{Pairing}
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%TODO Complete subsection
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Source : \citeannexes{wikipedia_ordered_pair}
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\langsubsubsection{Définition de Wiener}{Wiener's definition}
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$(a, b) := \{\{\{a\}, \emptyset\}, \{b\}\}$
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\langsubsubsection{Définition de Hausdorff}{Hausdorff's definition}
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$(a, b) := \{\{a, 1\}, \{b, 2\}\}$ where $a \ne 1 \land b \ne 2$
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\langsubsubsection{Définition de Kuratowski}{Kuratowski's definition}
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\begin{definition_sq} \label{definition:ordered_pair}
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$(a, b)_K := \{\{a\}, \{a, b\}\}$
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\end{definition_sq}
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\langsubsection{Réunion}{Union}
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Unite all elements of two given sets into one.
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\begin{definition_sq} \label{definition:set_union}
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$A \union B := \{x \suchthat (x \in A \lor x \in B)\}$
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\end{definition_sq}
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Pour des ensembles finis : $\forall (E, F) \in \Cat(\Set)^2, \card{E \union F} = \card{E} + \card{F} - \card{E \intersection F}$
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Example :
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$n,m \in \N$
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$A := \{a_0, \cdots, a_n\}$
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$B := \{b_0, \cdots, b_m\}$
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$A \union B = \{a_0, \cdots, a_n, b_0, \cdots, b_m\}$
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\langsubsection{Schéma de compréhension}{Scheme of replacement}
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%TODO Complete subsection
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\langsubsection{Infini}{Infinity}
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%TODO Complete subsection
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\subsection{Power set}
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%TODO Complete subsection
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For a set $S$ such that $\card{S} = n \implies \card{\mathbf{P}(S)} = 2^n$
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\langsubsection{Choix}{Choice}
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%TODO Complete subsection
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\begin{definition_sq} \label{definition:set_axiom_of_choice}
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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.
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$$\forall X [\emptyset \notin X \implies \exists \function{f}{X}{\Union_{A \in X} A \quad \forall A \in X(f(A) \in A)}]$$
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\end{definition_sq}
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\begin{theorem_sq} \label{theorem:ac_implies_lem}
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The axiom of choice implies the law of excluding middle.
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\end{theorem_sq}
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\begin{proof}
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Assume that $0 \ne 1$ (or any two elements that are not equal), Let $\Omega := \{0, 1\}$, $p \in \mathbf{Prop}$
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$A := \{ x \in \Omega \suchthat x = 0 \lor p \}$
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$B := \{ y \in \Omega \suchthat y = 1 \lor p \}$
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$\implies 0 \in A \land 1 \in B$
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$X := \{ A, B \}$, by definition $\Union X = \Omega$
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By the axiom of choice $\implies \exists \function{f}{X}{\Omega}$
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Using this function there are 4 cases:
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\begin{enumerate}[(1)]
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\item $f(A) = f(B) = 0 \implies 0 \in B$ but $((0 = 1) \lor p \implies \top) \implies p$
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\item $f(A) = f(B) = 1$ Same reasoning as (1) $\implies p$ % TODO Replace with local labeling and reference
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\item $f(A) \neq f(B) = 0 \implies A \neq B$ but $p \implies A = B = \Omega$ (contrapositive of (1) and (2)) $\implies \lnot p$
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\item $f(A) \neq f(B) = 1$ Same reasoning as (3) $\implies \lnot p$
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\end{enumerate}
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So by proof by cases $(p \lor \lnot p)$ which is the law of excluded middle \ref{definition:law_excluding_middle}.
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\end{proof}
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\section{Intersection}
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Unite all common elements of two given sets into one.
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\begin{definition_sq} \label{definition:set_intersection}
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$A \intersection B := \{x \suchthat (x \in A \land x \in B)\}$
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\end{definition_sq}
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Pour des ensembles finis : $\forall (E, F) \in \Set^2, \card{E \intersection F} = \card{E} - \card{F} + \card{E \union F}$
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Example :
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$n,m \in \N$
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$A := \{a_0, \cdots, a_n, c_0, \cdots, c_n\}$
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$B := \{b_0, \cdots, b_m, c_0, \cdots, c_n\}$
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$A \intersection B = \{c_0, \cdots, c_n\}$
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\langsection{Différence des sets}{Set difference}
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%TODO Complete section
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Exclude elements of a set from a set
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\begin{definition_sq} \label{definition:set_difference}
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$A \setminus B := \{x \suchthat (x \in A \land x \notin B)\}$
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\end{definition_sq}
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Pour des ensembles finis : $\forall (E, F) \in \Set^2, \card{E \setminus F} = \card{E} - \card{E \intersection F}$
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\langsection{Fonction}{Function}
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Source : \citeannexes{wikipedia_function_mathematics}
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\begin{definition_sq} \label{definition:set_function}
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Une fonction $f$ est un tuple d'un domaine \citeannexes{wikipedia_domain_function} $A$ et un codomaine \citeannexes{wikipedia_codomain} $B$.
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\end{definition_sq}
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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}.
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\subsection{Notation}
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$\functiondef{A}{B}$
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$\function{f}{x}{f(x)}$
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\langsubsection{Injectivité}{Injectivity}
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Source : \citeannexes{wikipedia_injective_function}
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\begin{definition_sq} \label{definition:injective}
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Une fonction $f$ de $E$ dans $F$ est dite \textbf{injective} si, et seulement si, $\forall (a,b) \in E, f(a) = f(b) \Rightarrow a = b$.
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\end{definition_sq}
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\langsubsection{Surjectivité}{Surjectivity}
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Source : \citeannexes{wikipedia_surjective_function}
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\begin{definition_sq} \label{definition:surjective}
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Une fonction $f$ de $E$ dans $F$ est dite \textbf{surjective} si, et seulement si, $\forall y \in F, \exists x \in E : y = f(x)$.
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\end{definition_sq}
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\langsubsection{Bijectivité}{Bijectivity}
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Source : \citeannexes{wikipedia_bijection}
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\begin{definition_sq} \label{definition:bijection}
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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 $\forall y \in F, \exists! x \in E : y = f(x)$.
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\end{definition_sq}
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Every bijection is an isomorphism \ref{definition:isomorphism} applied on the category \ref{definition:category} of sets \ref{set_theory}.
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