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packages/macros.sty : Added convinences macros

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saundersp 2024-11-07 05:29:13 +01:00
parent 94cdd07e84
commit aafcec6a3e
5 changed files with 46 additions and 36 deletions
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8
contents/algebra.tex
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@ -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}

36
contents/number_theory.tex
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@ -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}

12
contents/set_theory.tex
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@ -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}

10
contents/topology.tex
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@ -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}

16
packages/macros.sty
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@ -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}}