From aafcec6a3eb70392d1e2c4209108514592033a17 Mon Sep 17 00:00:00 2001 From: saundersp Date: Thu, 7 Nov 2024 05:29:13 +0100 Subject: [PATCH] packages/macros.sty : Added convinences macros --- contents/algebra.tex | 8 ++++---- contents/number_theory.tex | 36 +++++++++++++++++------------------- contents/set_theory.tex | 12 ++++++------ contents/topology.tex | 10 +++++----- packages/macros.sty | 16 ++++++++++++++-- 5 files changed, 46 insertions(+), 36 deletions(-) diff --git a/contents/algebra.tex b/contents/algebra.tex index 964532b..16f3cde 100644 --- a/contents/algebra.tex +++ b/contents/algebra.tex @@ -56,11 +56,11 @@ Un matrice est une structure qui permet de regrouper plusieurs éléments d'un c \subsection{Trace} %TODO Complete subsection -$\forall A \in \mathcal{M}_{n}, tr(A)=\sum_{k=0}^na_{kk}$ +$\forall A \in \mathcal{M}_{n}, tr(A)=\sum\limits_{k=0}^na_{kk}$ $tr\in\mathcal{L}(\mathcal{M}_n(\K),\K)$ -$\forall(A,B)\in\mathcal{M}_{n,p}(\K)\times\mathcal{M}_{p,n}(\K), tr(AB) = tr(BA)$ +$\forall(A,B)\in\mathcal{M}_{n,p}(\K)\cartesianProduct\mathcal{M}_{p,n}(\K), tr(AB) = tr(BA)$ \langsubsection{Valeurs propres}{Eigenvalues} %TODO Complete subsection @@ -80,12 +80,12 @@ $Eigenvalues = m \pm \sqrt{m^2-det(A)}$ \langsubsection{Déterminant}{Determinant} %%TODO Complete subsection -$\function{D}{\mathcal{M}_{m\times n}(\R)}{R}$ +$\function{D}{\mathcal{M}_{m\cartesianProduct n}(\R)}{R}$ \langsubsubsection{Axiomes}{Axioms} %%TODO Complete subsubsection -$\forall M \in \mathcal{M}_{m\times n}$ +$\forall M \in \mathcal{M}_{m\cartesianProduct n}$ \begin{itemize} \item{$\forall \lambda \in \K, D(\lambda M) = \lambda D(M)$} \end{itemize} diff --git a/contents/number_theory.tex b/contents/number_theory.tex index 6f3ba3c..193df5a 100644 --- a/contents/number_theory.tex +++ b/contents/number_theory.tex @@ -94,7 +94,7 @@ Il existe toujours un élément minimum pour n'importe quel sous-ensemble de $\N \langsection{Construction des entiers relatifs $(\Z)$}{Construction of relative numbers} %TODO Complete section -$\Z := \{\dots,-3,-2,-1,0,1,2,3,\dots\}$ +$\Z := \{\dots,-3,-2,-1,0,1,2,3,\dots\} = \Union_{n \in \N} n \union \Union_{n \in \N^*} -n$ \langsubsection{Relations binaries}{Binary relations} %TODO Complete subsection @@ -191,7 +191,7 @@ $i^2 = -1$ \begin{tabular}{|c||c|c|} \hline - & 1 & i \\ + $\cartesianProduct$ & 1 & i \\ \hline \hline 1 & 1 & i \\ @@ -229,7 +229,7 @@ Source: \citeannexes{wikipedia_quaternion} \begin{tabular}{|c||c|c|c|c|} \hline - & 1 & i & j & k \\ + $\cartesianProduct$ & 1 & i & j & k \\ \hline \hline 1 & 1 & i & j & k \\ @@ -251,7 +251,7 @@ Source: \citeannexes{wikipedia_octonion} \begin{tabular}{|c||c|c|c|c|c|c|c|c|} \hline - $e_i/e_j $ & $e_0$ & $e_1$ & $e_2$ & $e_3$ & $e_4$ & $e_5$ & $e_6$ & $e_7$ \\ + $\cartesianProduct$ & $e_0$ & $e_1$ & $e_2$ & $e_3$ & $e_4$ & $e_5$ & $e_6$ & $e_7$ \\ \hline \hline $e_0$ & $e_0$ & $e_1$ & $e_2$ & $e_3$ & $e_4$ & $e_5$ & $e_6$ & $e_7$ \\ @@ -289,7 +289,7 @@ Source: \citeannexes{wikipedia_sedenion} \begin{tabular}{|c|c|c|c|} \hline - & i & j & k \\ + $\cartesianProduct$ & i & j & k \\ \hline i & -1 & k & -j \\ \hline @@ -324,15 +324,13 @@ $\Pn = \{p | p \in \N^* \land p \text{ est premier}\} = p_0, p_1, \dots p_{n-1}, $\omega = (\prod_{p\in \Pn} p) + 1$ -$\forall p \in \Pn, \lnot(\omega \div p)$ +$\implies \forall p \in \Pn$, $\lnot(p \divides \omega)$ -$\omega \notin \Pn \land \omega \in \Pn$ +$\implies (\omega \notin \Pn \land \omega \in \Pn) \implies \bot$ -$\rightarrow\leftarrow$ +$\implies \card{P} = \infty$ -$\implies |P| = \infty$ - -Il existe une infinité de nombre premiers. +\end{proof} \langsubsection{Irrationnalité}{Irrationality} @@ -351,22 +349,22 @@ By contradiction let's assume $\sqrt{p} \in \Q$ $a \in \Z, b \in \N^*, \text{PGCD}(a,b) = 1, \sqrt{p} = \frac{a}{b}$ -$\Rightarrow p = (\frac{a}{b})^2 = \frac{a^2}{b^2}$ +$\implies p = (\frac{a}{b})^2 = \frac{a^2}{b^2}$ -$\Rightarrow b^2p = a^2$ +$\implies b^2p = a^2$ -$\Rightarrow p|a$ +$\implies p \divides a$ Let $c \in \N^*$, $a = pc$ -$\Rightarrow b^2 p = (pc)^2=p^2c^2$ +$\implies b^2 p = (pc)^2=p^2c^2$ -$\Rightarrow b^2 = pc^2$ +$\implies b^2 = pc^2$ -$\Rightarrow p|b$ +$\implies p \divides b$ -$\Rightarrow (p|b \land p|a \land \text{PGCD}(a,b)=1) \Rightarrow \bot$ +$\implies (p \divides b \land p \divides a \land \text{PGCD}(a,b)=1) \implies \bot$ -$\Rightarrow \sqrt{p} \notin \Q$ +$\implies \sqrt{p} \notin \Q$ \end{proof} diff --git a/contents/set_theory.tex b/contents/set_theory.tex index b5316a4..52884b5 100644 --- a/contents/set_theory.tex +++ b/contents/set_theory.tex @@ -13,7 +13,7 @@ $S = \{a,b,c\}$ \langsubsection{Extensionnalité}{Extensionality} -$\forall A\forall B(\forall X(X \in A \Leftrightarrow X \in B) \Rightarrow A = B)$ +$\forall A\forall B(\forall X(X \in A \equivalence X \in B) \implies A = B)$ \langsubsection{Spécification}{Specification} %TODO Complete subsection @@ -32,9 +32,9 @@ Unite all elements of two given sets into one. $n,m \in \N$ -$A = \{a_0, \cdots, a_n\}$ +$A := \{a_0, \cdots, a_n\}$ -$B = \{b_0, \cdots, b_m\}$ +$B := \{b_0, \cdots, b_m\}$ $A \union B = \{a_0, \cdots, a_n, b_0, \cdots, b_m\}$ @@ -47,7 +47,7 @@ $A \union B = \{a_0, \cdots, a_n, b_0, \cdots, b_m\}$ \subsection{Power set} %TODO Complete subsection -For a set $S$ such that $\card{S} = n \equivalence \card{\mathbf{P}(S)} = 2^n$ +For a set $S$ such that $\card{S} = n \implies \card{\mathbf{P}(S)} = 2^n$ \langsubsection{Choix}{Choice} %TODO Complete subsection @@ -77,9 +77,9 @@ If the domain is the same as the codomain then the function is an endormorphsim \subsection{Notation} -$A \longrightarrow B$ +$\functiondef{A}{B}$ -$ x \longrightarrow f(x)$ +$\function{f}{x}{f(x)}$ \langsubsection{Injectivité}{Injectivity} \label{definition:injective} diff --git a/contents/topology.tex b/contents/topology.tex index 1649250..119c281 100644 --- a/contents/topology.tex +++ b/contents/topology.tex @@ -33,8 +33,8 @@ On appellera $(E,\norm{.})$ un \textbf{espace vectoriel normé}. $n \in \N^*, E = \R^n$ \begin{itemize} - \item{$\norm{x}_1 = \sum_{i=0}^n \abs{x_i}$} - \item{$\norm{x}_2 = \sqrt{\sum_{i=0}^n x^2_i}$} + \item{$\norm{x}_1 = \sum\limits_{i=0}^n \abs{x_i}$} + \item{$\norm{x}_2 = \sqrt{\sum\limits_{i=0}^n x^2_i}$} \item{$\norm{x}_\infty = \max\{\abs{x_0}, \dots, \abs{x_n}\}$} \item{$E = R_n[X], \norm{P} = \int_0^1 \abs{P(x)}dx$} \item{$m \in \N^*, E = \mathcal{L}(R^n, R^m), \norm{\phi} = \max\{\norm{\phi(e_i)}_\infty, i \subseteq N^*\}$} ($e_i :=$ base canonique de $\R^n$) @@ -100,7 +100,7 @@ Soit $(E, \norm{.})$ un espace vectoriel normé et \suite{x} une suite d’élé Montrer que toute sous-suite de $(x_n)_{n \in \N}$ converge vers $l$. \\ -Soit $\epsilon > 0$, comme $\lim_{n \to +\infty} x_n = l$ +Soit $\epsilon > 0$, comme $\lim\limits_{n \to +\infty} x_n = l$ $\implies \exists n_0 \in \N$ tel que $\forall x \ge n_0$, $x_n \in \mathbb{B}(l, \epsilon)$ \\ @@ -156,7 +156,7 @@ $K$ est compact $\implies K$ possède un point d'accumulation. $K$ est compact \\ -Soit $\epsilon > 0$ \&\& $X = \{x_n, \forall n \in \N \}$ \&\& $X \subset K$ +Soit $\epsilon > 0 \land X = \{x_n, \forall n \in \N \} \land X \subset K$ $\implies \exists l \in K$ tel que $\lim\limits_{n \to +\infty} x_n = l \in \mathbb{B}(l, \epsilon) \subset K$ @@ -170,7 +170,7 @@ $\implies K$ possède un point d'accumulation $K$ possède un point d'accumulation. $\implies K$ est compact. \end{lemme_sq} -Soit $X = \{x_n, \forall n \in \N \}$ \&\& $X \subset K$ +Soit $X = \{x_n, \forall n \in \N \} \land X \subset K$ \paragraph{Si $X$ est fini} diff --git a/packages/macros.sty b/packages/macros.sty index a4ff967..1390b63 100644 --- a/packages/macros.sty +++ b/packages/macros.sty @@ -16,6 +16,7 @@ \newcommand{\R}{\mathbb{R}} % Real numbers symbol \newcommand{\C}{\mathbb{C}} % Complex numbers symbol \newcommand{\Cat}{\mathcal{C}} % Category +\newcommand{\Set}{\mathbf{Set}} % Set category \newcommand{\K}{\mathbb{K}} % Corps \newcommand{\Hq}{\mathbb{H}} % Quaternions numbers symbol \newcommand{\Ot}{\mathbb{O}} % Octonions numbers symbol @@ -24,6 +25,7 @@ \newcommand{\false}{F} % New symbol for false value \newcommand{\true}{V} % New symbol for true value \DeclareMathOperator{\Rel}{\mathcal{R}} % New symbol for binary relations +\DeclareMathOperator{\composes}{\circ} % New symbol composing morphisms \DeclarePairedDelimiter{\abs}{|}{|} \DeclarePairedDelimiter{\card}{|}{|} \DeclarePairedDelimiter{\floor}{\lfloor}{\rfloor} @@ -33,10 +35,14 @@ \newtheorem{theorem}{\lang{Théorème}{Theoreme}} \newtheorem{lemme}{Lemme} \newcommandx{\suite}[3][1=n,2=n]{$(#3_{#1})_{#2 \in \N}$} +\newcommand{\innerproduct}[2]{\langle #1, #2 \rangle} \newenvironment{definition_sq}{\begin{mdframed}\begin{definition}}{\end{definition}\end{mdframed}} \newenvironment{theorem_sq}{\begin{mdframed}\begin{theorem}}{\end{theorem}\end{mdframed}} \newenvironment{lemme_sq}{\begin{mdframed}\begin{lemme}}{\end{lemme}\end{mdframed}} -\newcommand{\norm}[1]{\|#1\|} +\newcommand{\norm}[1]{\lVert#1\rVert} +\newcommand{\Norm}[1]{\lVert #1\rVert} +\newcommand{\powerset}[1]{\mathcal{P}(#1)} % Power set +\newcommand{\converges}{\rightarrow} \newcommand{\equivalence}{\Leftrightarrow} \renewcommand{\implies}{\Longrightarrow} \newcommand{\Limplies}{\Longleftarrow} @@ -44,10 +50,16 @@ \newcommand{\Limpliespart}{\fbox{$\Limplies$}} \DeclareMathOperator{\divides}{\mid} \DeclareMathOperator{\suchas}{\text{\lang{tel que}{such as}}} -\renewcommand{\function}[3]{#1 : #2 \longrightarrow #3} +\renewcommand{\function}[3]{#1 \colon #2 \longrightarrow #3} \newcommand{\functiondef}[2]{\hspace{15pt}#1 \longmapsto #2} \newcommand{\otherwise}{\text{\lang{Sinon}{Otherwise}}} \DeclareMathOperator{\union}{\cup} +\DeclareMathOperator{\Union}{\bigcup} +\DeclareMathOperator{\intersection}{\cap} +\DeclareMathOperator{\Intersection}{\bigcap} +\DeclareMathOperator{\cartesianProduct}{\times} +\DeclareMathOperator{\CartesianProduct}{\bigtimes} +\newcommand{\discreteInterval}[1]{[\![#1]\!]} \renewcommand{\smallskip}{\vspace{3pt}} \renewcommand{\medskip}{\vspace{6pt}}