From 53c9776dd658f82165944a4d80a1056b55a0dfb0 Mon Sep 17 00:00:00 2001 From: Niklas Date: Sun, 25 Sep 2022 20:06:35 +0200 Subject: [PATCH] Add content to 2_abbildungen.md --- .../mathematik/mengenlehre/eindeutige_rel.tex | 72 ++++++++++++++++++ .../mengenlehre/relationen/2_abbildungen.md | 63 ++++++++++++++- .../mengenlehre/relationen/eindeutige_rel.png | Bin 0 -> 63892 bytes 3 files changed, 134 insertions(+), 1 deletion(-) create mode 100644 assets/mathematik/mengenlehre/eindeutige_rel.tex create mode 100644 static/img/mathematik/mengenlehre/relationen/eindeutige_rel.png diff --git a/assets/mathematik/mengenlehre/eindeutige_rel.tex b/assets/mathematik/mengenlehre/eindeutige_rel.tex new file mode 100644 index 0000000..3f17fdd --- /dev/null +++ b/assets/mathematik/mengenlehre/eindeutige_rel.tex @@ -0,0 +1,72 @@ +\documentclass[convert={size=512}]{standalone} +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{amsmath} +\usepackage{amssymb} +\usepackage{tikz} +\usepackage{color} + +\definecolor{draculaPurple}{RGB}{189,147,249} +\definecolor{draculaCyan}{RGB}{139,233,253} +\definecolor{draculaForeground}{RGB}{248,248,248} +\definecolor{draculaRed}{RGB}{255, 85, 85} + +\begin{document} + \begin{tikzpicture}[text=draculaForeground, draw=draculaCyan] + % Set names + \node at (0,-0.4) [thick] {A}; + \node at (3,-0.4) [thick] {B}; + + % Set elements + \begin{scope}[every node/.style={circle,thick,draw}] + \node (1) at (0,-1) {1}; + \node (2) at (0,-2) {2}; + \node (3) at (0,-3) {3}; + \node (4) at (0,-4) {4}; + \node (5) at (0,-5) {5}; + \end{scope} + \begin{scope}[every node/.style={circle,thick,draw}] + \node (A) at (3,-1.5) {a}; + \node (B) at (3,-2.5) {b}; + \node (C) at (3,-3.5) {c}; + \node (D) at (3,-4.5) {d}; + \end{scope} + + % Arrows + \begin{scope}[every edge/.style={draw=draculaPurple,very thick}] + \path [->] (1) edge (A); + \path [->] (1) edge (D); + \path [->] (2) edge (B); + \path [->] (3) edge (B); + \path [->] (4) edge (D); + \end{scope} + + % Set names + \node at (6,-0.4) [thick] {A}; + \node at (9,-0.4) [thick] {B}; + + % Set elements + \begin{scope}[every node/.style={circle,thick,draw}] + \node (11) at (6,-1) {1}; + \node (22) at (6,-2) {2}; + \node (33) at (6,-3) {3}; + \node (44) at (6,-4) {4}; + \node (55) at (6,-5) {5}; + \end{scope} + \begin{scope}[every node/.style={circle,thick,draw}] + \node (AA) at (9,-1.5) {a}; + \node (BB) at (9,-2.5) {b}; + \node (CC) at (9,-3.5) {c}; + \node (DD) at (9,-4.5) {d}; + \end{scope} + + % Arrows + \begin{scope}[every edge/.style={draw=draculaPurple,very thick}] + \path [->] (11) edge (AA); + \path [->] (22) edge (BB); + \path [->] (33) edge (BB); + \path [->] (44) edge (DD); + \end{scope} + + \end{tikzpicture} +\end{document} \ No newline at end of file diff --git a/docs/mathematik/mengenlehre/relationen/2_abbildungen.md b/docs/mathematik/mengenlehre/relationen/2_abbildungen.md index 6341838..81dc5aa 100644 --- a/docs/mathematik/mengenlehre/relationen/2_abbildungen.md +++ b/docs/mathematik/mengenlehre/relationen/2_abbildungen.md @@ -4,4 +4,65 @@ tags: [mathematik, mengenlehre, relation, abbildung, funktion] sidebar_position: 3 --- -WIP \ No newline at end of file +Eine der wichtigsten Relationen überhaupt stellen die *Abbildungen* da. +Sie tauchen überall in der Mathematik auf und in der Realität lassen sich viele Probleme durch diese +speziellen Relationen darstellen. + +## Funktionen +Bevor wir uns den Abbildungen widmen, definieren wir die Eindeutigkeit: + +:::note Eindeutigkeit + +Eine Relation $R \subseteq A \times B$ ist *eindeutig*, wenn +$$ + \forall x \in A\ \forall y, y' \in B:\ (x,y) \in R \wedge (x,y') \in R \rightarrow y = y' +$$ + +::: +Anschaulich gesprochen heißt das, dass von jedem $x$, das in der Relation $R$ vorkommt, nur ein Pfeil weggeht. +Gegeben seien $A = \{1, 2, 3, 4, 5\}$, $B = \{a, b, c, d\}$ und $R = \{(1,a), (1,d), (2,b), (3,b), (4,d)\} \subseteq A \times B$. +$R$ ist nicht eindeutig, da $(1,a)$ und $(1,d)$ vorkommen. +Laut Definition muss aber $y=y'$ gelten, wenn $x$ in $R$ enthalten, aber es gilt $a \ne d$. +Durch entfernen einer dieser Komponenten, z.B. $(1,d)$ wird $R$ eindeutig. +Im folgenden Schaubild ist links das ursprüngliche $R$ zu sehen, bei dem von der $1$ zwei Pfeile abgehen. +Rechts ist dann die eindeutige Relation, da von der linken Seite maximal ein Pfeil pro Element abgeht. +Bei der Eindeutigkeit ist es egal, wie viele Pfeile rechts auf ein Element zugehen, es geht nur um die linke Seite. + +![Eindeutigkeit bei einer Relation dargestellt](/img/mathematik/mengenlehre/relationen/eindeutige_rel.png) + +:::note Abbildung / Funktion + +$f$ ist *Abbildung* oder *Funktion* von $A$ in $B$ $:=$ +- $f$ ist Relation, also $f \subseteq A \times B$ +- $D(f) = A$, also $f$ ist Relation *__von__ $A$ in $B$* +- $f$ ist eindeutig + +::: +Man schreibt dafür: $f:\ A \rightarrow B$. +Statt $R(x) = \{y\}$ wie bei normalen Relationen, schreibt man verkürzt $f(x) = y$, da das $y$ ja nun eindeutig ist. + +Eine Funktion ordnet also jedem Element aus $A$ genau ein $y$ aus $B$ zu. + +In der Analysis und auch in anderen mathematischen Gebieten findet man häufig eine alternative Einführung: + +:::note Alternative Definition + +$f:\ A \rightarrow B$ ist Funktion $:=$ +$$ + \forall x \in A\ \exists ! y \in B:\ f(x) = y +$$ + +::: +Häufig wirds sie so eingeführt, ohne direkt auf das kartesische Produkt und Relationen zu verweisen. +In diesen Fällen wird dann der *Graph der Funktion* definiert als $G_f := \{(x,f(x))\ |\ x \in A \wedge f(x) \in B\} \subseteq A \times B$. +Aus mengentheoretischer Sicht stimmen also die Begriffe "Funktion" und "Graph der Funktion" in diesen Fällen überein. + +## Grundlegende Eigenschaften von Funktionen +Die *Verkettung*, oder auch *Komposition*, $f \circ g: A \rightarrow C$ von Funktionen $f:\ A \rightarrow B$ und $g:\ B \rightarrow C$ ist wieder eine Funktion. +Jedoch ist das *Inverse* $f^{-1}$ einer Funktion $f$ im Allgemeinen keine Funktion mehr. + +Zwei Funktionen $f$ und $g$ sind gleich $f=g$, wenn für alle $x$ gilt $f(x) = g(x)$. + + + +## Operationen \ No newline at end of file diff --git a/static/img/mathematik/mengenlehre/relationen/eindeutige_rel.png b/static/img/mathematik/mengenlehre/relationen/eindeutige_rel.png new file mode 100644 index 0000000000000000000000000000000000000000..6fdcd2918ac399a6d28ca32e406ab718a1d51177 GIT binary patch literal 63892 zcmbq)WmHt%8}AUp(A^*<-67o}-AG6a2q-C?L(G6ksz`TZkkUQ00)uqJfPhF1QUeS# zm-l`D>wdi-?p|xwIcuLeJD&LMr%s}=p%w`tBOw3)Akop*Fa-dx?{Bdv$?@(lD-RpK z0RRNt9_s4G9?s4HfOg`?k8Z{vtm#Hh&J^|`DwJvo#_I3*xuP@3pYcwAu2%V7V65r+ zrVU%*?HftWbWJQe+m9TtG^vS_VzFrh;smk3clgnms9O7OFce&|D0g9U!eKsW|J`+e z*tIVf+nYGo{vYAbO*{Hl#!?yRsmF0Qc$14Vp{M5oI2&`F6o9XT;T;>W>slbiI}g-O zTU*!bYrfb3K=GmL{J5@>HN5PHK3N(D>Sk$GZ@VeJVFmvU3O&@D>=nFuzkM)a7ezw| zVCK7FBE`gG{7>GR%fy~9R*!mC8of3Tv&d$9pVKl;@G$GgIrSs0A#RN*GX&-%ilsv9 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