From 1a17854c3c08a83df41d5bdb3d893bba674b716a Mon Sep 17 00:00:00 2001 From: saundersp Date: Mon, 5 Aug 2024 00:35:07 +0200 Subject: [PATCH] Fixed some notations mistakes --- contents/latex.tex | 2 +- contents/number_theory.tex | 10 +++++----- contents/set_theory.tex | 38 ++++++++++++++++++++++++++++---------- contents/topology.tex | 2 +- 4 files changed, 35 insertions(+), 17 deletions(-) diff --git a/contents/latex.tex b/contents/latex.tex index b67ea81..9b45fac 100644 --- a/contents/latex.tex +++ b/contents/latex.tex @@ -126,7 +126,7 @@ $\begin{Vmatrix} a & b \\ c & d \end{Vmatrix}$ \langsection{Informatique}{Computer science} %TODO Complete section -\subsection{LaTex} +\subsection{LaTeX} \begin{verbatim} \begin{verbatim} diff --git a/contents/number_theory.tex b/contents/number_theory.tex index 20b974e..0f41cfa 100644 --- a/contents/number_theory.tex +++ b/contents/number_theory.tex @@ -139,9 +139,9 @@ $\forall (p,q) \in \Q, \forall n \in \N^*, \frac{p}{q} \Leftrightarrow \frac{p \ \langsubsection{Opérateurs}{Operators} %TODO Complete subsection -$\forall ((p,q), (a,b)) \in \Q^2, \frac{p}{q} + \frac{a}{b} = \frac{pb + aq}{qb}$ +$\forall ((p,q), (a,b)) \in \Q, \frac{p}{q} + \frac{a}{b} = \frac{pb + aq}{qb}$ -$\forall ((p,q), (a,b)) \in \Q^2, \frac{p}{q} \cdot \frac{a}{b} = \frac{pa}{qb}$ +$\forall ((p,q), (a,b)) \in \Q, \frac{p}{q} \cdot \frac{a}{b} = \frac{pa}{qb}$ $\forall (p,q) \in \Q, \forall k \in \Z, (\frac{p}{q})^k = \frac{p^k}{q^k}$ @@ -183,7 +183,7 @@ $\functiondef{(p,q)}{P_1^{\frac{p}{|p|} + 1}P_2^pP_3^q}$ \citeannexes{wikipedia_complex_numbers} -$\C = (a,b) \in R^2, a + ib ~= \R^2 $ +$\C = (a,b) \in R, a + ib ~= \R $ $i^2 = -1$ @@ -204,7 +204,7 @@ $i^2 = -1$ \langsubsection{Relations binaries}{Binary relations} %TODO Complete subsection -$\forall ((a,b), (c,d)) \in \C^2, a = c \land b = d \Leftrightarrow a + ib = c + id$ +$\forall ((a,b), (c,d)) \in \C, a = c \land b = d \Leftrightarrow a + ib = c + id$ \langsubsection{Opérateurs}{Operators} %TODO Complete subsection @@ -213,7 +213,7 @@ Il est impossible d'avoir une relation d'ordre dans le corps des complexes mais \subsubsection{Ordre lexicographique} -$\forall((a,b),(c,d)) \in \C^2, a + ib \Rel_L c + id := \begin{cases} +$\forall((a,b),(c,d)) \in \C, a + ib \Rel_L c + id := \begin{cases} a < c & \implies a + ib < c + id \\ \otherwise & \begin{cases} b < d & \implies a + ib < c + id \\ diff --git a/contents/set_theory.tex b/contents/set_theory.tex index 2121d85..235d2b2 100644 --- a/contents/set_theory.tex +++ b/contents/set_theory.tex @@ -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, $\forall (a,b) \in E, f(a) = f(b) \Rightarrow a = b$. diff --git a/contents/topology.tex b/contents/topology.tex index 49b7635..2d0d38d 100644 --- a/contents/topology.tex +++ b/contents/topology.tex @@ -130,7 +130,7 @@ Toute sous-suites (ou suites extraite) d'un suite convergente vers $l \in E$ con Montrer que l’ensemble $\{x_n, n \in \N\}$ est borné. \\ -Sachant que $(x_n) \ in E$ converge vers $l \in E$ \&\& $\epsilon > 0$. +Sachant que $(x_n) \in E$ converge vers $l \in E \land \epsilon > 0$. $\Leftrightarrow \exists y \in E$ tel que $\{\forall n \in \N, x_n, l\} \subset \bar{\mathbb{B}}(y, \epsilon) \subset E$.