103 lines
3.0 KiB
TeX
103 lines
3.0 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 mathématiques 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 notée $\emptyset$.
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\langsubsection{Paire}{Pairing}
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%TODO Complete subsection
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\langsubsection{Réunion}{Union}
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%TODO Complete section
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Unite all elements of two given sets into one.
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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{Scheme of replacement}{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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\section{Intersection}
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Unite all common elements of two given sets into one.
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$n,m,i \in \N$
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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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$A \cap B = \{c_0, \cdots, c_n\}$
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\langsection{Différence des sets}{Set difference}
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%TODO Complete section
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\langsection{Fonction}{Function}
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Source: \citeannexes{wikipedia_function_mathematics}
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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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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}.
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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} \label{definition:injective}
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Source: \citeannexes{wikipedia_injective_function}
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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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\langsubsection{Surjectivité}{Surjectivity} \label{definition:surjective}
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Source: \citeannexes{wikipedia_surjective_function}
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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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\langsubsection{Bijectivité}{Bijectivity}
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Source: \citeannexes{wikipedia_bijection} \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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Every bijection is an isomorphism \ref{definition:isomorphism} applied on set theory \ref{set_theory}.
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