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\emph{assoziativ}. + + Sei $f: X \rightarrow Y$ eine Funktion: + \begin{enumerate}[i.] + \item $f $ ist injektiv $\lrarr \exists g: Y \rightarrow X$, sodass $g \circ f = id_x$ + \item $f$ ist surjektiv $\lrarr \exists g: Y \rightarrow X$, sodass $f \circ g = id_y$ + \item $f$ ist bijektiv $\lrarr \exists g: Y \rightarrow X$, sodass $g \circ f = id_x, f \circ g = id_y$ + \end{enumerate} + \section{Kombinatorik} \subsection{Binomialkoeffizient}\label{kombi:binomial} Der \href{https://de.wikipedia.org/wiki/Binomialkoeffizient}{Binomialkoeffizient} diff --git a/MafIA1/mafia.pdf b/MafIA1/mafia.pdf index dd6b7777acda457e1c73d49ab1333a229a92c2b7..eb2f5998bccd1b695d7efd535bafb4791664239c 100644 GIT binary patch delta 63999 zcmZU)Q(z^~!lc`=ZQFKsjE-%qW82uVZQHhOCmq|i-N~GD?>x*u^R`~s%U4yaYMthS zPc4I^P$>QuXJBFEfTcLPyMBXZV@*Jz0J3wj2?)SCyEvH|*}{5kTx)g3(s99GKVm$B zRY@-4G3$a$M!JBs!YQv^Ufy87|4=4VUvTJqyWQ%S;m||UT?#87?Taov+!kU+0Ilxp zJ+Q>Wr#4@Y+g6WP1(!g@-}aUty!5)!Sx1jYwDxQC^_hS~#ZUUu?gL20s`kp6KsH}o 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Ein $y$ kann also durch mehrere $x$ getroffen werden, es gibt jedoch kein $y$, zu dem es keinen $x$-Wert gibt. + \subsubsection{Beweise} + Nachfolgend betrachten wir $f: \R \rightarrow \R, x \mapsto mx+b, m\ne 0$.\\ + Um die \emph{Injektivität} einer Funktion zu beweisen, nehmen wir + die umgekehrte Definition, also $f(x) = f(y) \Rightarrow x = y$. + \begin{proof} + \emph{Zu zeigen:} $f(x) = f(y) \Rightarrow x = y$\\ + Sei $f(x) = f(y)$ mit $x,y \in \R$ beliebig. + \begin{align*} + f(x) &= f(y)\\ + mx+b &= my+b\\ + mx &= my\\ + x &= y + \end{align*} + Da $x=y$, ist $f$ injektiv. + \end{proof} + + Um \emph{Surjektivität} zu zeigen, wird zunächst die Definition von + $x$ ermittelt: + \begin{align*} + f(x) &= y \\ + mx+b &= y \\ + mx &= y - b \\ + x &= \frac{y-b}{m} + \end{align*} + Diese Definition macht man sich nun im Beweis zu nutze, um $f(x) = y$ + für beliebige $y$ zu zeigen: + \begin{proof} + Sei $y \in \R$ beliebig. Aus vorheriger Berechnung ist bekannt: + $x = \frac{y-b}{m}$ + \begin{align*} + f(x) = f\left(\frac{y-b}{m}\right) = m\cdot \frac{y-b}{m} + b + = y - b + b = y + \end{align*} + Daraus resultiert, dass $f$ surjektiv ist. + \end{proof} + Da $f$ surjektiv und injektiv ist, folgt auch die \emph{Surjektivität} + für $f$. + +\section{Zahlen}\label{zahlen} + \subsection{Natürliche Zahlen} + \subsubsection{Peano-Axiome} + Definition der natürlichen Zahlen durch Peano: + \begin{enumerate} + \item $0 \in \N$ + \item es gibt eine Nachfolgerabbildung $succ: \N \rightarrow \N \backslash \{0\}$ + \item $succ$ ist injektiv + \item Ist $M \subseteq \N$ mit + \begin{enumerate}[i.] + \item $0 \in M$ + \item $m \in M \Rightarrow succ(m) \in M \forall m \in M$ + \end{enumerate} + so gilt $M= \N$. + \end{enumerate} + + \subsection{Gruppen}\label{zahlen:gruppen} + Eine nichtleere Menge $G$ mit einer Verknüpfung $\circ$ heißt Gruppe, + wenn die folgenden Eigenschaften zutreffen: + \begin{enumerate} + \item Assoziativität von $\circ$, also $(a \circ b) \circ c = a \circ (b \circ c) \; \forall a,b,c \in G$ + \item Es existiert ein neutrales Element $e$, für das gilt: $e \in G: a \circ e =a \; \forall a \in G$ + \item Zu jedem Element gibt es ein Inverses $a^{-1}$, für das gilt: $a \circ a^{-1} = e$ + \end{enumerate} + Gilt darüber hinaus die \emph{Kommutativität} + \[a \circ b = b \circ a \forall a,b \in G \] + heißt die Gruppe abelsch oder kommutativ. + \section{Vektorräume}\label{vektorraum} \subsection{Vektorraum}\label{vektorraum:def} Eine nichtleere Menge $V$ heißt Vektorraum über einem Körper $\K$, wenn die folgenden Eigenschaften zutreffen:\\