saundersp/notebook
1
0
Fork 0
Code Issues Pull Requests Packages Projects Releases Wiki Activity

Compare commits

base: saundersp:d0c9dab3300d186601ee338f05caed9399c5b1ae
Branches Tags
saundersp:main
..
compare: saundersp:c8b77b243593469506a50ef10e41a027e3d38ccd
Branches Tags
saundersp:main

No commits in common. "d0c9dab3300d186601ee338f05caed9399c5b1ae" and "c8b77b243593469506a50ef10e41a027e3d38ccd" have entirely different histories.
d0c9dab330 ... c8b77b2435

10 changed files with 111 additions and 317 deletions
Show Stats Expand all files Collapse all files

4
Dockerfile
View File

@ -1,9 +1,9 @@
FROM alpine:3.20.3 FROM alpine:3.20.2
RUN apk add --no-cache \ RUN apk add --no-cache \
make=4.4.1-r2 \ make=4.4.1-r2 \
graphviz=9.0.0-r2 \ graphviz=9.0.0-r2 \
texlive-xetex=20240210.69778-r4 \ texlive-xetex=20240210.69778-r3 \
texmf-dist-langfrench=2024.0-r5 \ texmf-dist-langfrench=2024.0-r5 \
texmf-dist-latexextra=2024.0-r5 \ texmf-dist-latexextra=2024.0-r5 \
texmf-dist-bibtexextra=2024.0-r5 \ texmf-dist-bibtexextra=2024.0-r5 \

213
contents/algebra.tex
View File

@ -6,19 +6,19 @@
\subsection{Magma} \label{definition:magma} \subsection{Magma} \label{definition:magma}
Soit un ensemble $S$ avec une loi de composition interne $(\star)$ notée $(S,\star)$ tel que $\forall(a,b) \in S, a \star b \in S$. Soit une structure $S$ avec une loi de composition interne $(+)$ notée $(S,+)$ tel que $\forall(a,b) \in S, a + b \in S$.
\langsubsection{Magma unital}{Unital magma} \label{definition:unital_magma} \langsubsection{Magma unital}{Unital magma} \label{definition:unital_magma}
Soit un magma \ref{definition:magma} $(S,\star)$ untial en $0_e$ tel que $\exists 0_e \in S, \forall a \in S, 0_e \star a = a$. Soit un magma \ref{definition:magma} $(S,+)$ untial en $0_e$ tel que $\exists 0_e \in S, \forall a \in S, 0_e + a = a$.
\subsection{Monoïd} \label{definition:monoid} \subsection{Monoïd} \label{definition:monoid}
Soit un magma unital \ref{definition:unital_magma} $(S,\star)$ dont la loi de composition est associative \ref{definition:associativity}. Soit un magma unital \ref{definition:unital_magma} $(S,+)$ dont la loi de composition est associative \ref{definition:associativity}.
\langsubsection{Groupe}{Group} \label{definition:group} \langsubsection{Groupe}{Group} \label{definition:group}
Soit un monoïd \ref{definition:monoid} $(G,\star)$ ayant un élément inverse tel que $\forall a \in G, \exists a^{-1} \in G, a \star a^{-1} = 0_e$. Soit un monoid \ref{definition:monoid} $(G,+)$ ayant un élément inverse tel que $\forall a \in G, \exists a^{-1} \in G, a + a^{-1} = 0_e$.
\langsubsubsection{Groupe abélien}{Abelian group} \label{definition:abelian_group} \langsubsubsection{Groupe abélien}{Abelian group} \label{definition:abelian_group}
@ -26,17 +26,13 @@ Un groupe abélien est un groupe \ref{definition:group} dont la loi de compositi
\langsubsection{Corps}{Field} \label{definition:field} \langsubsection{Corps}{Field} \label{definition:field}
Soit une structure $F$ avec deux lois de composition interne $(+)$ et $(\cartesianProduct)$ notée $(F,+,\cartesianProduct)$. Soit une structure $F$ avec deux lois de composition interne $(+)$ et $(\times)$ notée $(F,+,\times)$.
\begin{itemize} \begin{itemize}
\item{$(F,+)$ est un groupe \ref{definition:group} unital en $0_e$} \item{$(F,+)$ est un groupe \ref{definition:group} unital en $0_e$}
\item{$(F\backslash\{0_e\},\cartesianProduct)$ est un groupe \ref{definition:group}} \item{$(F\backslash\{0_e\},\times)$ est un groupe \ref{definition:group}}
\end{itemize} \end{itemize}
\langsubsubsection{Corps abélien}{Abelian field} \label{definition:abelian_field}
Un corps abélien est un corps \ref{definition:field} dont la loi de composition est commutatif \ref{definition:commutativity}.
\langsubsection{Anneau}{Ring} \label{definition:ring} \langsubsection{Anneau}{Ring} \label{definition:ring}
%TODO Complete subsection %TODO Complete subsection
@ -56,11 +52,11 @@ Un matrice est une structure qui permet de regrouper plusieurs éléments d'un c
\subsection{Trace} \subsection{Trace}
%TODO Complete subsection %TODO Complete subsection
$\forall A \in \mathcal{M}_{n}, tr(A)=\sum\limits_{k=0}^na_{kk}$ $\forall A \in \mathcal{M}_{n}, tr(A)=\sum_{k=0}^na_{kk}$
$tr\in\mathcal{L}(\mathcal{M}_n(\K),\K)$ $tr\in\mathcal{L}(\mathcal{M}_n(\K),\K)$
$\forall(A,B)\in\mathcal{M}_{n,p}(\K)\cartesianProduct\mathcal{M}_{p,n}(\K), tr(AB) = tr(BA)$ $\forall(A,B)\in\mathcal{M}_{n,p}(\K)\times\mathcal{M}_{p,n}(\K), tr(AB) = tr(BA)$
\langsubsection{Valeurs propres}{Eigenvalues} \langsubsection{Valeurs propres}{Eigenvalues}
%TODO Complete subsection %TODO Complete subsection
@ -80,12 +76,12 @@ $Eigenvalues = m \pm \sqrt{m^2-det(A)}$
\langsubsection{Déterminant}{Determinant} \langsubsection{Déterminant}{Determinant}
%%TODO Complete subsection %%TODO Complete subsection
$\function{D}{\mathcal{M}_{m\cartesianProduct n}(\R)}{R}$ $\function{D}{\mathcal{M}_{m\times n}(\R)}{R}$
\langsubsubsection{Axiomes}{Axioms} \langsubsubsection{Axiomes}{Axioms}
%%TODO Complete subsubsection %%TODO Complete subsubsection
$\forall M \in \mathcal{M}_{m\cartesianProduct n}$ $\forall M \in \mathcal{M}_{m\times n}$
\begin{itemize} \begin{itemize}
\item{$\forall \lambda \in \K, D(\lambda M) = \lambda D(M)$} \item{$\forall \lambda \in \K, D(\lambda M) = \lambda D(M)$}
\end{itemize} \end{itemize}
@ -190,37 +186,6 @@ Et vérifiant $\forall(\alpha,\beta) \in \K, \forall(a,b,c) \in E$
\item{Distributivité (gauche et droite) $*$ de $\K \Leftrightarrow a(\alpha*\beta)+(\alpha*\beta)a+\alpha(\beta a)$} \item{Distributivité (gauche et droite) $*$ de $\K \Leftrightarrow a(\alpha*\beta)+(\alpha*\beta)a+\alpha(\beta a)$}
\end{itemize} \end{itemize}
\langsubsection{Famille libre}{Free family} \label{definition:vector_space_free_family}
\begin{definition_sq}
Une famille \suite{e} est dite \textbf{libre} si
$$\forall i \in \discreteInterval{1, n}, \lambda_i \in K, \sum\limits_{i = 1}^n \lambda_i e_i = 0 \implies \lambda_i = 0$$
\end{definition_sq}
\langsubsection{Famille génératrice}{Generating family} \label{definition:vector_space_generating_family}
\begin{definition_sq}
Une famille \suite{e} est dite \textbf{génératrice} de $E$ si
$$\forall v \in E, \exists \lambda \in K^n, \sum\limits_{i=1}^n \lambda_i e_i = v$$
\end{definition_sq}
\langsubsection{Bases}{Basis} \label{definition:vector_space_basis}
\begin{definition_sq}
Une famille est dite une \textbf{base} de $E$ si elle est libre \ref{definition:vector_space_free_family} et génératrice \ref{definition:vector_space_generating_family} $\equivalence \forall v \in E, \exists! \lambda \in K^n, \sum\limits_{i=1}^n \lambda_i e_i = v$
\end{definition_sq}
\subsection{Dimension} \label{definition:vector_space_dimension}
%TODO Complete subsection
\langsubsubsection{Rang}{Rank} \label{definition:vector_space_rank}
%TODO Complete subsubsection
\begin{theorem_sq} \label{theorem:vector_space_rank}
Soit $E$ et $G$ $K$-e.v \ref{definition:sub_vector_space} et $\function{\phi}{E}{F}$.
$\dim E = \dim \ker(\phi) + \dim im(\phi) = \dim \ker(\phi) = rg(\phi)$
\end{theorem_sq}
\langsubsection{Sous-espaces vectoriels}{Sub vector spaces} \label{definition:sub_vector_space} \langsubsection{Sous-espaces vectoriels}{Sub vector spaces} \label{definition:sub_vector_space}
%TODO Complete subsection %TODO Complete subsection
@ -266,161 +231,3 @@ $\implies F \subset G \lor G \subset F$
\end{proof} \end{proof}
\langsubsection{Application linéaire}{Linear maps} \label{definition:linearity}
Une application linéaire est un morphisme \ref{definition:morphism}
appliqué à la catégorie \ref{definition:category}
des espaces vectoriels \ref{definition:vector_space}.
\langsubsubsection{Axiomes}{Axioms}
Given $f: \mathbb{K} \rightarrow \mathbb{K}$
\begin{itemize}
\item{Additivity: $\forall(x,y) \in \mathbb{K}, f(x+y)=f(x)+f(y)$}
\item{Homogeneity: $\forall(a,x) \in \mathbb{K}, f(ax)=af(x)$}
\item{Or (a faster way): $\forall(a,x,y) \in \mathbb{K}, f(x + ay) = f(x) + af(y)$}
\end{itemize}
\langsubsection{Forme bilinéaire}{Bilinear form} \label{definition:bilinear_form}
\langsubsubsection{Axiomes}{Axioms}
Une forme bilinéaire est une fonction $\function{B}{E \cartesianProduct E}{K}$ sur un $\K$-espace vectoriel $E$ qui est linéaire sur les deux arguments tel qui respectes les axiomes suivants :
\begin{itemize}
\item{$\forall u,v,w \in E, B(u + v,w) = B(u,w) + B(v,w)$}
\item{$\forall u,v \in E, \forall a \in K, B(au,w) = aB(u,w)$}
\item{$\forall u,v,w \in E, B(u,w + v) = B(u,v) + B(u,w)$}
\item{$\forall u,v \in E, \forall a \in K, B(u,aw) = aB(u,w)$}
\end{itemize}
\langsubsection{Produit scalaire}{Inner product}
\langsubsubsection{Produit scalaire réel}{Real inner product}
\langsubsubsubsection{Axiomes}{Axioms}
Un produit scalaire notée $\innerproduct{-}{-}$ sur un $\R$-espace vectoriel $E$ est une forme bilinéaire \ref{definition:bilinear_form} qui respectes les axiomes suivants :
\begin{itemize}
\item{Symétrie: $\forall(x,y) \in E, \innerproduct{x}{y} = \innerproduct{y}{x}$}
\item{Non-dégénérescence: $\forall x \in E, \innerproduct{x}{x} = 0 \implies x = 0$}
\end{itemize}
\langsubsection{Norme Réel}{Real norm}
\langsubsubsection{Axiomes}{Axioms}
Une norme notée $\norm{.}_E$ sur un $\R$-espace vectoriel $E$ est une application $\function{\norm{.}}{E}{R_+}$ qui respectes les axiomes suivants :
\begin{itemize}
\item{Séparation: $\forall x \in E, \norm{x} = 0 \implies x = 0$}
\item{Homogénéité: $\forall x \in E, \forall \lambda \in \R \norm{\lambda x} = \abs{\lambda}\norm{x}$}
\item{Inégalité triangulaire: $\forall x,y \in E, \norm{x + y} \le \norm{x} + \norm{y}$}
\end{itemize}
\langsubsection{Espace pré-hilbertien}{Pre-hilbertian Space}
A $\K$-espace vectoriel muni d'un produit scalaire est appelé un espace pré-hilbertien.
\langsubsection{Espace Euclidien}{Euclidian Space}
Un espace euclidien est une espace pré-hilbertien réel à dimension finie.
\pagebreak
\section{Devoir Maison 1 : Algèbre multilinéaire}
\section{Exercice 1}
Soit $(E,\innerproduct{.}{.})$ un espace euclidien. On définit
$$\function{i}{E \setminus \{0\}}{E \setminus \{0\}}$$
$$\functiondef{x}{\frac{x}{\norm{x}^2}}$$
qu'on appelle \textit{inversion} de centre 0 et de rapport 1.
\begin{enumerate}
\item{Montrer que $i$ est une bijection de $E \setminus \{0\}$ sur lui-même, vérifiant $i \composes i = id_E$}
\begin{proof}\par
Si $i$ est une bijection de $E$ alors il existe une fonction réciproque (ou inverse) $i^{-1}$ telle que $i \composes i^{-1} = id_E$, or $i$ est défini comme son propre inverse. Donc il suffit d'évaluer $i$ avec lui-même pour terminer la preuve.
$$i \composes i = \frac{\frac{x}{\norm{x}^2}}{\norm{\frac{x}{\norm{x}^2}}^2} = \frac{\frac{x}{\norm{x}^2}}{\frac{\norm{x}^2}{\norm{x}^4}} = \frac{\norm{x}^2 x}{\norm{x}^2} = x = id_E$$
\end{proof}
\item{Montrer $$\forall x,y \in E \setminus \{0\}, \frac{\innerproduct{i(x)}{i(y)}}{\norm{i(x)}\norm{i(y)}} = \frac{\innerproduct{x}{y}}{\norm{x}\norm{y}}$$ On dit que $i$ est une application \textit{conforme}.}
\begin{proof}\par
Soit $x,y \in E \setminus \{0\}$
$$\frac{\innerproduct{i(x)}{i(y)}}{\norm{i(x)}\norm{i(y)}} = \frac{\innerproduct{\frac{x}{\norm{x}^2}}{\frac{y}{\norm{y}^2}}}{\norm{\frac{x}{\norm{x}^2}}\norm{\frac{y}{\norm{y}^2}}} = \frac{\frac{\innerproduct{x}{y}}{\norm{x}^2\norm{y}^2}}{\frac{\norm{x}\norm{y}}{\norm{x}^2\norm{y}^2}} = \frac{\innerproduct{x}{y}}{\norm{x}\norm{y}}$$
\end{proof}
\item{Démontrer que $$\forall x,y \in E \setminus \{0\}, \norm{i(x) - i(y)} = \frac{\norm{x - y}}{\norm{x}\norm{y}}$$}
\begin{proof}\par
Soit $x,y \in E \setminus \{0\}$
$$\norm{i(x) - i(y)} = \norm{\frac{x}{\norm{x}^2} - \frac{y}{\norm{y}^2}} = \norm{\frac{\norm{y}^2 x - \norm{x}^2 y}{\norm{x}^2\norm{y}^2}} = \frac{\norm{\norm{y}^2 x - \norm{x}^2 y}}{\norm{x}^2\norm{y}^2} = \frac{\sqrt{\innerproduct{\norm{y}^2 x - \norm{x}^2 y}{\norm{y}^2 x - \norm{x}^2 y}}}{\norm{x}^2\norm{y}^2}$$
$$= \frac{\sqrt{\innerproduct{\norm{y}^2 x}{\norm{y}^2 x} - 2 \innerproduct{\norm{y}^2 x}{\norm{x}^2 y} - \innerproduct{\norm{x}^2 y}{\norm{x}^2 y}}}{\norm{x}^2\norm{y}^2} = \frac{\sqrt{\norm{y}^4 \norm{x}^2 - 2\norm{y}^2 \norm{x}^2 \innerproduct{x}{y} - \norm{x}^4\norm{y}^2}}{\norm{x}^2\norm{y}^2}$$
$$= \frac{\sqrt{\norm{y}^2 \norm{x}^2 (\norm{y}^2 - 2 \innerproduct{x}{y} - \norm{x}^2)}}{\norm{x}^2\norm{y}^2} = \frac{\norm{x}\norm{y}\sqrt{\innerproduct{x - y}{x - y}}}{\norm{x}^2\norm{y}^2} = \frac{\norm{x - y}}{\norm{x}\norm{y}}$$
\end{proof}
\item{En déduire que pour tous $x,y,z \in E \setminus \{0\}$, on a $$\norm{x}\norm{y - z} \le \norm{y}\norm{x - z} + \norm{z}\norm{x - y}$$}
\begin{proof}\par
Posons $a,b \in E \setminus \{0\}$ tel que $a := i(y) - i(x)$ et $b := i(x) - i(z)$, puis utilisons l'inégalité triangulaire $\norm{a + b} \le \norm{a} + \norm{b}$ et développons.
$$\norm{(i(y) - i(x)) + (i(x) - i(z))} = \norm{i(y) - i(z)} \le \norm{i(x) - i(z)} + \norm{i(x) - i(y)}$$
Par le résultat de (3).
$$\frac{\norm{y - z}}{\norm{y}\norm{z}} \le \frac{\norm{x - z}}{\norm{x}\norm{z}} + \frac{\norm{x - y}}{\norm{x}\norm{y}}$$
En multipliant par $\norm{x}\norm{y}\norm{z}$
$$\norm{x}\norm{y - z} \le \norm{y}\norm{x - z} + \norm{z}\norm{x - y}$$
\end{proof}
\item{En déduire que pour tous $a,b,c,d \in E \setminus \{0\}$, on a $$\norm{a - c}\norm{b - d} \le \norm{a - b}\norm{c - d} + \norm{a - d}\norm{b - c}$$ C'est l'\textit{inégalité de Ptolémée}.}
\bigskip
Pour cette preuve nous aurons besoin de ce lemme :
\begin{lemme_sq} \label{norm_diff_symetry}
$\forall (e,f) \in E, \norm{e - f} = \norm{f - e}$
\begin{proof}\par
Soit $E$ un $\K$-espace vectoriel et soit $e,f \in E$.
Comme $\exists (-1_K) \in \K(+, \cartesianProduct) \suchas (-1_K) \cartesianProduct (-1_K) = 1_K$.
$$\norm{e - f} = \norm{-1_\K(f - e)} = \abs{-1_\K}\norm{f - e} = \norm{f - e}$$
\end{proof}
\end{lemme_sq}
\begin{proof}\par
Soit $a,b,c,d \in E$.
Comme $E$ est un espace vectoriel et donc un groupe par $E(+)$.
Posons $x,y,z \in E$ tel que $x := a - c$, $y := a - b$ et $z := a - d$.
Ainsi, par le résultat (4).
$$\norm{x}\norm{y - z} \le \norm{y}\norm{x - z} + \norm{z}\norm{x - y}$$
Par le lemme (\ref{norm_diff_symetry}).
$$\norm{x}\norm{z - y} \le \norm{y}\norm{z - x} + \norm{z}\norm{y - x}$$
En développant $x$, $y$ et $z$ on obtient
$$\norm{a - c}\norm{(a - d) - (a - b)} \le \norm{a - b}\norm{(a - d) - (a - c)} + \norm{a - d}\norm{(a - b) - (a - c)}$$
$$\norm{a - c}\norm{b - d} \le \norm{a - b}\norm{c - d} + \norm{a - d}\norm{b - c}$$
\end{proof}
Soit $u \in E$ tel que $\norm{u} = 1$ et soit $\alpha \ne 0$. Considérons $H = \{ x \in E | \innerproduct{x}{u} = \alpha \}$. C'est un hyperplan affine de $E$.
\item{Justifier que $0 \notin H$. En utilisant la question 3, montrer alors que $i(H) = S \setminus \{0\}$, où $$S = \lbrace x \in E | \Norm{x - \frac{1}{2\alpha}u} = \frac{1}{2\abs{\alpha}} \rbrace$$}
% TODO Complete 6.
Soient $a \in E$ et $R > 0$. On note $S(a,R) = \{ x \in E | \norm{x - a} = R\}$ la sphère de centre $a$ et de rayon $R$.
\item{On suppose que $\norm{a} \ne R$. Montrer que $0 \notin S(a,R)$ et que $$ i(S(a,R)) = S(\frac{a}{\norm{a}^2 - R^2}, \frac{R}{\abs{\norm{a}^2 - R^2}})$$}
% TODO Complete 7.
\end{enumerate}

36
contents/number_theory.tex
View File

@ -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} \langsection{Construction des entiers relatifs $(\Z)$}{Construction of relative numbers}
%TODO Complete section %TODO Complete section
$\Z := \{\dots,-3,-2,-1,0,1,2,3,\dots\} = \Union_{n \in \N} n \union \Union_{n \in \N^*} -n$ $\Z := \{\dots,-3,-2,-1,0,1,2,3,\dots\}$
\langsubsection{Relations binaries}{Binary relations} \langsubsection{Relations binaries}{Binary relations}
%TODO Complete subsection %TODO Complete subsection
@ -191,7 +191,7 @@ $i^2 = -1$
\begin{tabular}{|c||c|c|} \begin{tabular}{|c||c|c|}
\hline \hline
$\cartesianProduct$ & 1 & i \\ & 1 & i \\
\hline \hline
\hline \hline
1 & 1 & i \\ 1 & 1 & i \\
@ -229,7 +229,7 @@ Source: \citeannexes{wikipedia_quaternion}
\begin{tabular}{|c||c|c|c|c|} \begin{tabular}{|c||c|c|c|c|}
\hline \hline
$\cartesianProduct$ & 1 & i & j & k \\ & 1 & i & j & k \\
\hline \hline
\hline \hline
1 & 1 & i & j & k \\ 1 & 1 & i & j & k \\
@ -251,7 +251,7 @@ Source: \citeannexes{wikipedia_octonion}
\begin{tabular}{|c||c|c|c|c|c|c|c|c|} \begin{tabular}{|c||c|c|c|c|c|c|c|c|}
\hline \hline
$\cartesianProduct$ & $e_0$ & $e_1$ & $e_2$ & $e_3$ & $e_4$ & $e_5$ & $e_6$ & $e_7$ \\ $e_i/e_j $ & $e_0$ & $e_1$ & $e_2$ & $e_3$ & $e_4$ & $e_5$ & $e_6$ & $e_7$ \\
\hline \hline
\hline \hline
$e_0$ & $e_0$ & $e_1$ & $e_2$ & $e_3$ & $e_4$ & $e_5$ & $e_6$ & $e_7$ \\ $e_0$ & $e_0$ & $e_1$ & $e_2$ & $e_3$ & $e_4$ & $e_5$ & $e_6$ & $e_7$ \\
@ -289,7 +289,7 @@ Source: \citeannexes{wikipedia_sedenion}
\begin{tabular}{|c|c|c|c|} \begin{tabular}{|c|c|c|c|}
\hline \hline
$\cartesianProduct$ & i & j & k \\ & i & j & k \\
\hline \hline
i & -1 & k & -j \\ i & -1 & k & -j \\
\hline \hline
@ -324,13 +324,15 @@ $\Pn = \{p | p \in \N^* \land p \text{ est premier}\} = p_0, p_1, \dots p_{n-1},
$\omega = (\prod_{p\in \Pn} p) + 1$ $\omega = (\prod_{p\in \Pn} p) + 1$
$\implies \forall p \in \Pn$, $\lnot(p \divides \omega)$ $\forall p \in \Pn, \lnot(\omega \div p)$
$\implies (\omega \notin \Pn \land \omega \in \Pn) \implies \bot$ $\omega \notin \Pn \land \omega \in \Pn$
$\implies \card{P} = \infty$ $\rightarrow\leftarrow$
\end{proof} $\implies |P| = \infty$
Il existe une infinité de nombre premiers.
\langsubsection{Irrationnalité}{Irrationality} \langsubsection{Irrationnalité}{Irrationality}
@ -349,22 +351,22 @@ By contradiction let's assume $\sqrt{p} \in \Q$
$a \in \Z, b \in \N^*, \text{PGCD}(a,b) = 1, \sqrt{p} = \frac{a}{b}$ $a \in \Z, b \in \N^*, \text{PGCD}(a,b) = 1, \sqrt{p} = \frac{a}{b}$
$\implies p = (\frac{a}{b})^2 = \frac{a^2}{b^2}$ $\Rightarrow p = (\frac{a}{b})^2 = \frac{a^2}{b^2}$
$\implies b^2p = a^2$ $\Rightarrow b^2p = a^2$
$\implies p \divides a$ $\Rightarrow p|a$
Let $c \in \N^*$, $a = pc$ Let $c \in \N^*$, $a = pc$
$\implies b^2 p = (pc)^2=p^2c^2$ $\Rightarrow b^2 p = (pc)^2=p^2c^2$
$\implies b^2 = pc^2$ $\Rightarrow b^2 = pc^2$
$\implies p \divides b$ $\Rightarrow p|b$
$\implies (p \divides b \land p \divides a \land \text{PGCD}(a,b)=1) \implies \bot$ $\Rightarrow (p|b \land p|a \land \text{PGCD}(a,b)=1) \Rightarrow \bot$
$\implies \sqrt{p} \notin \Q$ $\Rightarrow \sqrt{p} \notin \Q$
\end{proof} \end{proof}

12
contents/set_theory.tex
View File

@ -13,7 +13,7 @@ $S = \{a,b,c\}$
\langsubsection{Extensionnalité}{Extensionality} \langsubsection{Extensionnalité}{Extensionality}
$\forall A\forall B(\forall X(X \in A \equivalence X \in B) \implies A = B)$ $\forall A\forall B(\forall X(X \in A \Leftrightarrow X \in B) \Rightarrow A = B)$
\langsubsection{Spécification}{Specification} \langsubsection{Spécification}{Specification}
%TODO Complete subsection %TODO Complete subsection
@ -32,9 +32,9 @@ Unite all elements of two given sets into one.
$n,m \in \N$ $n,m \in \N$
$A := \{a_0, \cdots, a_n\}$ $A = \{a_0, \cdots, a_n\}$
$B := \{b_0, \cdots, b_m\}$ $B = \{b_0, \cdots, b_m\}$
$A \union B = \{a_0, \cdots, a_n, b_0, \cdots, b_m\}$ $A \union B = \{a_0, \cdots, a_n, b_0, \cdots, b_m\}$
@ -47,7 +47,7 @@ $A \union B = \{a_0, \cdots, a_n, b_0, \cdots, b_m\}$
\subsection{Power set} \subsection{Power set}
%TODO Complete subsection %TODO Complete subsection
For a set $S$ such that $\card{S} = n \implies \card{\mathbf{P}(S)} = 2^n$ For a set $S$ such that $\card{S} = n \equivalence \card{\mathbf{P}(S)} = 2^n$
\langsubsection{Choix}{Choice} \langsubsection{Choix}{Choice}
%TODO Complete subsection %TODO Complete subsection
@ -77,9 +77,9 @@ If the domain is the same as the codomain then the function is an endormorphsim
\subsection{Notation} \subsection{Notation}
$\functiondef{A}{B}$ $A \longrightarrow B$
$\function{f}{x}{f(x)}$ $ x \longrightarrow f(x)$
\langsubsection{Injectivité}{Injectivity} \label{definition:injective} \langsubsection{Injectivité}{Injectivity} \label{definition:injective}

10
contents/topology.tex
View File

@ -33,8 +33,8 @@ On appellera $(E,\norm{.})$ un \textbf{espace vectoriel normé}.
$n \in \N^*, E = \R^n$ $n \in \N^*, E = \R^n$
\begin{itemize} \begin{itemize}
\item{$\norm{x}_1 = \sum\limits_{i=0}^n \abs{x_i}$} \item{$\norm{x}_1 = \sum_{i=0}^n \abs{x_i}$}
\item{$\norm{x}_2 = \sqrt{\sum\limits_{i=0}^n x^2_i}$} \item{$\norm{x}_2 = \sqrt{\sum_{i=0}^n x^2_i}$}
\item{$\norm{x}_\infty = \max\{\abs{x_0}, \dots, \abs{x_n}\}$} \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{$E = R_n[X], \norm{P} = \int_0^1 \abs{P(x)}dx$}
\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 $\R^n$) \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 $l$. Montrer que toute sous-suite de $(x_n)_{n \in \N}$ converge vers $l$.
\\ \\
Soit $\epsilon > 0$, comme $\lim\limits_{n \to +\infty} x_n = l$ Soit $\epsilon > 0$, comme $\lim_{n \to +\infty} x_n = l$
$\implies \exists n_0 \in \N$ tel que $\forall x \ge n_0$, $x_n \in \mathbb{B}(l, \epsilon)$ $\implies \exists n_0 \in \N$ 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 $K$ est compact
\\ \\
Soit $\epsilon > 0 \land X = \{x_n, \forall n \in \N \} \land X \subset K$ Soit $\epsilon > 0$ \&\& $X = \{x_n, \forall n \in \N \}$ \&\& $X \subset K$
$\implies \exists l \in K$ tel que $\lim\limits_{n \to +\infty} x_n = l \in \mathbb{B}(l, \epsilon) \subset K$ $\implies \exists l \in K$ 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. $K$ possède un point d'accumulation. $\implies K$ est compact.
\end{lemme_sq} \end{lemme_sq}
Soit $X = \{x_n, \forall n \in \N \} \land X \subset K$ Soit $X = \{x_n, \forall n \in \N \}$ \&\& $X \subset K$
\paragraph{Si $X$ est fini} \paragraph{Si $X$ est fini}

2
graphs/countable_integers.gv
View File

@ -1,4 +1,4 @@
digraph denumberabilityIntegers { digraph {
node [shape = plaintext, fontcolor = White, fontsize = 30]; node [shape = plaintext, fontcolor = White, fontsize = 30];
rankdir = LR; rankdir = LR;
bgcolor = None; bgcolor = None;

53
graphs/countable_rationals.gv
View File

@ -1,15 +1,13 @@
digraph denumberabilityRationals { digraph {
node [shape = plaintext, fontcolor = White, fontsize = 15]; node [shape = plaintext, fontcolor = White, fontsize = 30];
rankdir = LR; rankdir = LR;
bgcolor = None; bgcolor = None;
Edge [fontcolor = White, color = White, fontsize = 15]; Edge [fontcolor = White, color = White, fontsize = 25];
subgraph dots { subgraph dots {
node [label = "..."]; node [label = "..."];
d; d2; d3; d4; d5; d; d2; d3; d4; d5;
vd; vd2; vd3; vd4; vd5; vd6; vd; vd2; vd3; vd4; vd5; vd6;
vd5 -> d5;
vd -> vd2 -> vd3 -> vd4 -> vd5 -> vd6 [color = None];
} }
subgraph pos { subgraph pos {
@ -19,30 +17,26 @@ digraph denumberabilityRationals {
"2/1" -> "3/1" [taillabel = 2]; "2/1" -> "3/1" [taillabel = 2];
"3/1" -> "2/2" [taillabel = 3]; "3/1" -> "2/2" [taillabel = 3];
"2/2" -> "1/3" [taillabel = 4]; "2/2" -> "1/3" [taillabel = 4];
"1/3" -> "1/4" [taillabel = 5]; "1/3" -> "2/3" [taillabel = 5];
"1/4" -> "2/3" [taillabel = 6]; "2/3" -> "3/2" [taillabel = 6];
"2/3" -> "3/2" [taillabel = 7]; "3/2" -> "4/1" [taillabel = 7];
"3/2" -> "4/1" [taillabel = 8]; "4/1" -> "5/1" [taillabel = 8];
"4/1" -> "5/1" [taillabel = 9]; "5/1" -> "4/2" [taillabel = 9];
"5/1" -> "4/2" [taillabel = 10]; "4/2" -> "3/3" [taillabel = 10];
"4/2" -> "3/3" [taillabel = 11]; "3/3" -> "2/4" [taillabel = 11];
"3/3" -> "2/4" [taillabel = 12]; "2/4" -> "1/4" [taillabel = 12];
"2/4" -> "1/5" [taillabel = 13]; "1/4" -> "1/5" [taillabel = 13];
"1/5" -> vd [taillabel = 14]; "1/5" -> "2/5" [taillabel = 14];
vd -> "2/5" [taillabel = 15]; //"2/5" -> "3/4" [taillabel = 15];
"2/5" -> "3/4" [taillabel = 16]; //"3/4" -> "4/3" [taillabel = 16];
"3/4" -> "4/3" [taillabel = 17]; "4/3" -> "5/2" [taillabel = 17];
"4/3" -> "5/2" [taillabel = 18];
"5/2" -> d [taillabel = 19]; "1/5" -> vd [color = None];
d2 -> "5/3" [taillabel = 22]; "2/5" -> vd2 [color = None];
"5/3" -> "4/4" [taillabel = 23]; "3/5" -> vd3 [color = None];
"4/4" -> "3/5" [taillabel = 24]; "4/5" -> vd4 [color = None];
"3/5" -> vd2 [taillabel = 25]; "5/5" -> vd5 [color = None];
vd3 -> "4/5" [taillabel = 31]; d5 -> vd6 [color = None];
"4/5" -> "5/4" [taillabel = 32];
"5/4" -> d3 [taillabel = 33];
d4 -> "5/5" [taillabel = 41];
"5/5" -> vd4 [taillabel = 42];
} }
"1/1" -> "2/1" -> "3/1" -> "4/1" -> "5/1" -> d [color = None]; "1/1" -> "2/1" -> "3/1" -> "4/1" -> "5/1" -> d [color = None];
@ -50,4 +44,5 @@ digraph denumberabilityRationals {
"1/3" -> "2/3" -> "3/3" -> "4/3" -> "5/3" -> d3 [color = None]; "1/3" -> "2/3" -> "3/3" -> "4/3" -> "5/3" -> d3 [color = None];
"1/4" -> "2/4" -> "3/4" -> "4/4" -> "5/4" -> d4 [color = None]; "1/4" -> "2/4" -> "3/4" -> "4/4" -> "5/4" -> d4 [color = None];
"1/5" -> "2/5" -> "3/5" -> "4/5" -> "5/5" -> d5 [color = None]; "1/5" -> "2/5" -> "3/5" -> "4/5" -> "5/5" -> d5 [color = None];
vd -> vd2 -> vd3 -> vd4 -> vd5 -> vd6 [color = None];
} }

10
main.tex
View File

@ -19,15 +19,15 @@
%\usepackage{lipsum} % Command to generate temporary dummy text %\usepackage{lipsum} % Command to generate temporary dummy text
\usepackage[ruled,vlined,linesnumbered]{algorithm2e} % Add the algorithm environnement \usepackage[ruled,vlined,linesnumbered]{algorithm2e} % Add the algorithm environnement
\usepackage[codedark]{packages/themes} % Include many colours themes ([default], codedark or dracula) \usepackage[codedark]{packages/themes} % Include many colours themes ([default], codedark or dracula)
\pagecolor{theme_colour_background} \pagecolor{th_colour_bg}
\color{theme_colour_foreground} \color{th_colour_fg}
\usepackage{amsmath} % Provides command to typeset matrices with different delimiters \usepackage{amsmath} % Provides command to typeset matrices with different delimiters
\usepackage{listings} % Add an environnement to highlight code \usepackage{listings} % Add an environnement to highlight code
\usepackage{xargs} % Allow multiple optional parameters parsing \usepackage{xargs} % Allow multiple optional parameters parsing
\usepackage{mdframed} % Fancy rectangles \usepackage{mdframed} % Fancy rectangles
\mdfsetup{linecolor = theme_colour_foreground, innerlinecolor = theme_colour_foreground, % \mdfsetup{linecolor = th_colour_fg, innerlinecolor = th_colour_fg,%
middlelinecolor = theme_colour_foreground, outerlinecolor = theme_colour_foreground, % middlelinecolor = th_colour_fg, outerlinecolor = th_colour_fg,%
backgroundcolor = theme_colour_background, fontcolor = theme_colour_foreground} backgroundcolor = th_colour_bg, fontcolor = th_colour_fg}
\usepackage{packages/macros} % Customs macros \usepackage{packages/macros} % Customs macros
\usepackage{graphicx} \usepackage{graphicx}
\usepackage{makeidx}[intoc] % Make a word index \usepackage{makeidx}[intoc] % Make a word index

22
packages/macros.sty
View File

@ -16,16 +16,16 @@
\newcommand{\R}{\mathbb{R}} % Real numbers symbol \newcommand{\R}{\mathbb{R}} % Real numbers symbol
\newcommand{\C}{\mathbb{C}} % Complex numbers symbol \newcommand{\C}{\mathbb{C}} % Complex numbers symbol
\newcommand{\Cat}{\mathcal{C}} % Category \newcommand{\Cat}{\mathcal{C}} % Category
\newcommand{\Set}{\mathbf{Set}} % Set category
\newcommand{\K}{\mathbb{K}} % Corps \newcommand{\K}{\mathbb{K}} % Corps
\newcommand{\Hq}{\mathbb{H}} % Quaternions numbers symbol \newcommand{\Hq}{\mathbb{H}} % Quaternions numbers symbol
\newcommand{\Ot}{\mathbb{O}} % Octonions numbers symbol \newcommand{\Ot}{\mathbb{O}} % Octonions numbers symbol
\newcommand{\Se}{\mathbb{S}} % Sedenions numbers symbol \newcommand{\Se}{\mathbb{S}} % Sedenions numbers symbol
\newcommand{\Pn}{\mathbb{P}} % Sets of all the prime numbers \newcommand{\Pn}{\mathbb{P}} % Sets of all the prime numbers
\newcommand{\false}{F} % New symbol for false value \newcommand{\false}{{\color{th_colour_red}F}} % New symbol for false value
\newcommand{\true}{V} % New symbol for true value \newcommand{\true}{{\color{th_colour_green}V}} % New symbol for true value
%\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{\Rel}{\mathcal{R}} % New symbol for binary relations
\DeclareMathOperator{\composes}{\circ} % New symbol composing morphisms
\DeclarePairedDelimiter{\abs}{|}{|} \DeclarePairedDelimiter{\abs}{|}{|}
\DeclarePairedDelimiter{\card}{|}{|} \DeclarePairedDelimiter{\card}{|}{|}
\DeclarePairedDelimiter{\floor}{\lfloor}{\rfloor} \DeclarePairedDelimiter{\floor}{\lfloor}{\rfloor}
@ -35,14 +35,10 @@
\newtheorem{theorem}{\lang{Théorème}{Theoreme}} \newtheorem{theorem}{\lang{Théorème}{Theoreme}}
\newtheorem{lemme}{Lemme} \newtheorem{lemme}{Lemme}
\newcommandx{\suite}[3][1=n,2=n]{$(#3_{#1})_{#2 \in \N}$} \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{definition_sq}{\begin{mdframed}\begin{definition}}{\end{definition}\end{mdframed}}
\newenvironment{theorem_sq}{\begin{mdframed}\begin{theorem}}{\end{theorem}\end{mdframed}} \newenvironment{theorem_sq}{\begin{mdframed}\begin{theorem}}{\end{theorem}\end{mdframed}}
\newenvironment{lemme_sq}{\begin{mdframed}\begin{lemme}}{\end{lemme}\end{mdframed}} \newenvironment{lemme_sq}{\begin{mdframed}\begin{lemme}}{\end{lemme}\end{mdframed}}
\newcommand{\norm}[1]{\lVert#1\rVert} \newcommand{\norm}[1]{\|#1\|}
\newcommand{\Norm}[1]{\lVert #1\rVert}
\newcommand{\powerset}[1]{\mathcal{P}(#1)} % Power set
\newcommand{\converges}{\rightarrow}
\newcommand{\equivalence}{\Leftrightarrow} \newcommand{\equivalence}{\Leftrightarrow}
\renewcommand{\implies}{\Longrightarrow} \renewcommand{\implies}{\Longrightarrow}
\newcommand{\Limplies}{\Longleftarrow} \newcommand{\Limplies}{\Longleftarrow}
@ -50,16 +46,10 @@
\newcommand{\Limpliespart}{\fbox{$\Limplies$}} \newcommand{\Limpliespart}{\fbox{$\Limplies$}}
\DeclareMathOperator{\divides}{\mid} \DeclareMathOperator{\divides}{\mid}
\DeclareMathOperator{\suchas}{\text{\lang{tel que}{such as}}} \DeclareMathOperator{\suchas}{\text{\lang{tel que}{such as}}}
\renewcommand{\function}[3]{#1 \colon #2 \longrightarrow #3} \renewcommand{\function}[3]{#1 : #2 \longrightarrow #3}
\newcommand{\functiondef}[2]{\hspace{15pt}#1 \longmapsto #2} \newcommand{\functiondef}[2]{\hspace{15pt}#1 \longmapsto #2}
\newcommand{\otherwise}{\text{\lang{Sinon}{Otherwise}}} \newcommand{\otherwise}{\text{\lang{Sinon}{Otherwise}}}
\DeclareMathOperator{\union}{\cup} \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{\smallskip}{\vspace{3pt}}
\renewcommand{\medskip}{\vspace{6pt}} \renewcommand{\medskip}{\vspace{6pt}}

66
packages/themes.sty
View File

@ -4,44 +4,44 @@
\RequirePackage{xcolor} \RequirePackage{xcolor}
\DeclareOption{default}{\OptionNotUsed} \DeclareOption{default}{\OptionNotUsed}
\definecolor{theme_colour_background} {RGB} {255, 255, 255} \definecolor{th_colour_bg} {RGB} {255, 255, 255}
\definecolor{theme_colour_foreground} {RGB} {0, 0, 0 } \definecolor{th_colour_fg} {RGB} {0, 0, 0 }
\definecolor{theme_colour_cl} {RGB} {68, 71, 90 } \definecolor{th_colour_cl} {RGB} {68, 71, 90 }
\definecolor{theme_colour_comment} {RGB} {98, 114, 164} \definecolor{th_colour_comment} {RGB} {98, 114, 164}
\definecolor{theme_colour_cyan} {RGB} {139, 233, 253} \definecolor{th_colour_cyan} {RGB} {139, 233, 253}
\definecolor{theme_colour_green} {RGB} {0, 255, 0 } \definecolor{th_colour_green} {RGB} {0, 255, 0 }
\definecolor{theme_colour_orange} {RGB} {255, 184, 108} \definecolor{th_colour_orange} {RGB} {255, 184, 108}
\definecolor{theme_colour_pink} {RGB} {255, 121, 198} \definecolor{th_colour_pink} {RGB} {255, 121, 198}
\definecolor{theme_colour_purple} {RGB} {189, 147, 249} \definecolor{th_colour_purple} {RGB} {189, 147, 249}
\definecolor{theme_colour_red} {RGB} {255, 0, 0 } \definecolor{th_colour_red} {RGB} {255, 0, 0 }
\definecolor{theme_colour_yellow} {RGB} {255, 255, 0 } \definecolor{th_colour_yellow} {RGB} {255, 255, 0 }
\DeclareOption{codedark}{ \DeclareOption{codedark}{
\definecolor{theme_colour_background} {HTML} {222324} \definecolor{th_colour_bg} {HTML} {222324}
\definecolor{theme_colour_foreground} {HTML} {FFFFFF} \definecolor{th_colour_fg} {HTML} {FFFFFF}
\definecolor{theme_colour_cl} {RGB} {68, 71, 90 } \definecolor{th_colour_cl} {RGB} {68, 71, 90 }
\definecolor{theme_colour_comment} {RGB} {98, 114, 164} \definecolor{th_colour_comment} {RGB} {98, 114, 164}
\definecolor{theme_colour_cyan} {RGB} {139, 233, 253} \definecolor{th_colour_cyan} {RGB} {139, 233, 253}
\definecolor{theme_colour_green} {RGB} {80, 250, 123} \definecolor{th_colour_green} {RGB} {80, 250, 123}
\definecolor{theme_colour_orange} {RGB} {255, 184, 108} \definecolor{th_colour_orange} {RGB} {255, 184, 108}
\definecolor{theme_colour_pink} {RGB} {255, 121, 198} \definecolor{th_colour_pink} {RGB} {255, 121, 198}
\definecolor{theme_colour_purple} {RGB} {189, 147, 249} \definecolor{th_colour_purple} {RGB} {189, 147, 249}
\definecolor{theme_colour_red} {RGB} {255, 85, 85 } \definecolor{th_colour_red} {RGB} {255, 85, 85 }
\definecolor{theme_colour_yellow} {RGB} {241, 250, 140} \definecolor{th_colour_yellow} {RGB} {241, 250, 140}
} }
\DeclareOption{dracula}{ \DeclareOption{dracula}{
\definecolor{theme_colour_background} {RGB} {40, 42, 54 } \definecolor{th_colour_bg} {RGB} {40, 42, 54 }
\definecolor{theme_colour_foreground} {RGB} {248, 248, 242} \definecolor{th_colour_fg} {RGB} {248, 248, 242}
\definecolor{theme_colour_cl} {RGB} {68, 71, 90 } \definecolor{th_colour_cl} {RGB} {68, 71, 90 }
\definecolor{theme_colour_comment} {RGB} {98, 114, 164} \definecolor{th_colour_comment} {RGB} {98, 114, 164}
\definecolor{theme_colour_cyan} {RGB} {139, 233, 253} \definecolor{th_colour_cyan} {RGB} {139, 233, 253}
\definecolor{theme_colour_green} {RGB} {80, 250, 123} \definecolor{th_colour_green} {RGB} {80, 250, 123}
\definecolor{theme_colour_orange} {RGB} {255, 184, 108} \definecolor{th_colour_orange} {RGB} {255, 184, 108}
\definecolor{theme_colour_pink} {RGB} {255, 121, 198} \definecolor{th_colour_pink} {RGB} {255, 121, 198}
\definecolor{theme_colour_purple} {RGB} {189, 147, 249} \definecolor{th_colour_purple} {RGB} {189, 147, 249}
\definecolor{theme_colour_red} {RGB} {255, 85, 85 } \definecolor{th_colour_red} {RGB} {255, 85, 85 }
\definecolor{theme_colour_yellow} {RGB} {241, 250, 140} \definecolor{th_colour_yellow} {RGB} {241, 250, 140}
} }
\ProcessOptions\relax \ProcessOptions\relax