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Fixed some notations mistakes

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saundersp
2024-08-05 00:35:07 +02:00
parent 55ac1c6c20
commit 1a17854c3c
4 changed files with 35 additions and 17 deletions
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38
contents/set_theory.tex
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@ -16,6 +16,10 @@ $\forall A\forall B(\forall X(X \in A \Leftrightarrow X \in B) \Rightarrow A = B
\langsubsection{Spécification}{Specification}
%TODO Complete subsection
\langsubsection{Ensemble vide}{Empty set}
Il existe un ensemble vide notée $\emptyset$.
\langsubsection{Paire}{Pairing}
%TODO Complete subsection
@ -24,13 +28,13 @@ $\forall A\forall B(\forall X(X \in A \Leftrightarrow X \in B) \Rightarrow A = B
Unite all elements of two given sets into one.
$n,m \in \N^+$
$n,m \in \N$
$A = \{a_1, \cdots, a_n\}$
$A = \{a_0, \cdots, a_n\}$
$B = \{b_1, \cdots, b_m\}$
$B = \{b_0, \cdots, b_m\}$
$A \cup B = \{a_1, \cdots, a_n, b_1, \cdots, b_m\}$
$A \union B = \{a_0, \cdots, a_n, b_0, \cdots, b_m\}$
\langsubsection{Scheme of replacement}{Scheme of replacement}
%TODO Complete subsection
@ -41,29 +45,43 @@ $A \cup B = \{a_1, \cdots, a_n, b_1, \cdots, b_m\}$
\subsection{Power set}
%TODO Complete subsection
For a set $S$ such that $|S| = n \Leftrightarrow \mathbf{P}(S) = 2^n$
\langsubsection{Choix}{Choice}
%TODO Complete subsection
\section{Intersection}
%TODO Complete subsection
Unite all common elements of two given sets into one.
$n,m,i \in \N$
$A = \{a_0, \cdots, a_n, c_0, \cdots, c_n\}$
$B = \{b_0, \cdots, b_m, c_0, \cdots, c_n\}$
$A \cap B = \{c_0, \cdots, c_n\}$
\langsection{Différence des sets}{Set difference}
%TODO Complete section
\langsection{Fonction}{Function}
%TODO Complete section
Une fonction $f$ est un opération qui permet de transformer un ou plusieurs éléments d'un set $A$ en d'autres éléments d'un set $B$.
Source: \citeannexes{wikipedia_function_mathematics}
Une fonction $f$ est un tuple d'un domaine \citeannexes{wikipedia_domain_function} $A$ et un codomaine \citeannexes{wikipedia_codomain} $B$.
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}.
\subsection{Notation}
%TODO Complete subsection
$A \longrightarrow B$
$ x \longrightarrow f(x)$
\langsubsection{Injectivité}{Injectivity}
%TODO Complete subsection
\langsubsection{Injectivité}{Injectivity} \label{definition:injective}
Source: \citeannexes{wikipedia_injective_function}
Une fonction $f$ de $E$ dans $F$ est dite \textbf{injective} si, et seulement si, $\forall (a,b) \in E, f(a) = f(b) \Rightarrow a = b$.