From 2d037730e0ec087b0470c47ec164ddb3e5384f31 Mon Sep 17 00:00:00 2001 From: Niklas Date: Mon, 3 Oct 2022 15:50:30 +0200 Subject: [PATCH] Add more content to 2_abbildungen.md - examples for functions - surjection, injection and bijection --- assets/mathematik/mengenlehre/bijektion.tex | 100 +++++++++++++++++ .../mengenlehre/relationen/2_abbildungen.md | 106 ++++++++++++++++-- .../mengenlehre/relationen/bijektion.png | Bin 0 -> 80487 bytes 3 files changed, 197 insertions(+), 9 deletions(-) create mode 100644 assets/mathematik/mengenlehre/bijektion.tex create mode 100644 static/img/mathematik/mengenlehre/relationen/bijektion.png diff --git a/assets/mathematik/mengenlehre/bijektion.tex b/assets/mathematik/mengenlehre/bijektion.tex new file mode 100644 index 0000000..6ead24c --- /dev/null +++ b/assets/mathematik/mengenlehre/bijektion.tex @@ -0,0 +1,100 @@ +\documentclass[convert={size=768}]{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] + % Injection + % Set names + \node at (1.5,0) [thick] {injektiv}; + \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}; + \end{scope} + \begin{scope}[every node/.style={circle,thick,draw}] + \node (A) at (3,-1) {a}; + \node (B) at (3,-2) {b}; + \node (C) at (3,-3) {c}; + \node (D) at (3,-4) {d}; + \end{scope} + + % Arrows + \begin{scope}[every edge/.style={draw=draculaPurple,very thick}] + \path [->] (1) edge (D); + \path [->] (2) edge (A); + \path [->] (3) edge (C); + \end{scope} + + % Surjection + % Set names + \node at (7.5,0) [thick] {surjektiv}; + \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) {a}; + \node (BB) at (9,-2) {b}; + \node (CC) at (9,-3) {c}; + \node (DD) at (9,-4) {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); + \path [->] (55) edge (CC); + \end{scope} + + % Bijection + % Set names + \node at (13.5,0) [thick] {bijektiv}; + \node at (12,-0.4) [thick] {A}; + \node at (15,-0.4) [thick] {B}; + + % Set elements + \begin{scope}[every node/.style={circle,thick,draw}] + \node (111) at (12,-1) {1}; + \node (222) at (12,-2) {2}; + \node (333) at (12,-3) {3}; + \node (444) at (12,-4) {4}; + \end{scope} + \begin{scope}[every node/.style={circle,thick,draw}] + \node (AAA) at (15,-1) {a}; + \node (BBB) at (15,-2) {b}; + \node (CCC) at (15,-3) {c}; + \node (DDD) at (15,-4) {d}; + \end{scope} + + % Arrows + \begin{scope}[every edge/.style={draw=draculaPurple,very thick}] + \path [->] (111) edge (DDD); + \path [->] (222) edge (AAA); + \path [->] (333) edge (CCC); + \path [->] (444) edge (BBB); + \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 81dc5aa..ae1e423 100644 --- a/docs/mathematik/mengenlehre/relationen/2_abbildungen.md +++ b/docs/mathematik/mengenlehre/relationen/2_abbildungen.md @@ -37,25 +37,62 @@ $f$ ist *Abbildung* oder *Funktion* von $A$ in $B$ $:=$ - $D(f) = A$, also $f$ ist Relation *__von__ $A$ in $B$* - $f$ ist eindeutig +Mathematisch-logisch aufgeschrieben: +$$ + \forall x \in A\ \exists ! y \in B:\ (x,y) \in A \times B +$$ + ::: 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: +In der Analysis und auch in anderen mathematischen Gebieten findet man häufig eine alternative textliche Einführung: :::note Alternative Definition -$f:\ A \rightarrow B$ ist Funktion $:=$ -$$ - \forall x \in A\ \exists ! y \in B:\ f(x) = y -$$ +Eine Abbildung oder eine Funktion einer Menge $A$ in eine Menge $B$ ordnet jedem Element $x \in A$ genau ein Element $y \in B$ zu. ::: -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. +Bei dieser Definition wird komplett auf die mengentheoretische Begriffe wie *geordnetes Paar* oder auch *Relation* verzichtet. +Erst aufbauend darauf wird dann mengentheoretisch 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 in diesen Fällen die Begriffe "Funktion" und "Graph der Funktion" überein. + +#### Beispiele +Gegeben seien $A = \{1, 2, 3, 4, 5\}$ und $B = \{a, b, c, d\}$. +Eine Abbildung $f:\ A \rightarrow B$ können dann sein: +- $f = \{(1,b)\}$ +- $f = \{(1,a), (2,b), (3,b), (4,d)\}$ +- $f = \{(1,b), (2,c), (3,d), (4,a), (5,b)\}$ +- ... + +Wenn man eine rechnerische Vorschrift (*Abbildungsvorschrift*) angeben möchte, wie ein $y$ zu einem $x$ berechnet werden soll, +dann schreibt man Abbildungen so: + +:::note Schreibweisen + +$$ + f:\ A \rightarrow B,\ x \mapsto f(x) + \qquad\qquad\text{oder}\qquad\qquad + f: \begin{cases} + A \rightarrow B\\ + x \mapsto f(x) + \end{cases} +$$ +wobei anstelle $f(x)$ dann z.B. eine *Funktionsgleichung* tritt. + +::: + +#### Beispiele +- $f:\ \mathbb{R} \rightarrow \mathbb{R},\ x \mapsto x^2 \quad$ für eine Parabel +- $f: \begin{cases} + \mathbb{N} \rightarrow \mathbb{R}\\ + x \mapsto \sqrt{x} + \end{cases} \quad$ für Wurzeln von natürlichen Zahlen + +Es werden noch mehr unterschiedliche Darstellungsformen für Funktionen auftauchen, je nachdem wie es gerade angebracht ist. +Bis dahin sollten einem die Begriffe allerdings so vertraut vorkommen, dass eine genaue Definition oder Einführung nicht notwendig ist. ## 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. @@ -63,6 +100,57 @@ Jedoch ist das *Inverse* $f^{-1}$ einer Funktion $f$ im Allgemeinen keine Funkti Zwei Funktionen $f$ und $g$ sind gleich $f=g$, wenn für alle $x$ gilt $f(x) = g(x)$. +:::note Injektivität, Surjektivität, Bijektivität +Sei $f: A \rightarrow B$ eine Funktion. +- $f$ heißt *injektiv* (*eineindeutig*) $:=\ \forall x,x' \in A:\ f(x) = f(x') \rightarrow x = x'$ +- $f$ heißt *surjektiv* (*Abbildung auf $B$*) $:=\ \forall y \in B\ \exists x \in A:\ f(x) = y$ +- $f$ heißt *bijektiv* (*eineindeutige Abbildung auf $B$*) $:=\ f$ ist injektiv und surjektiv -## Operationen \ No newline at end of file +::: +*Injektivität* bedeutet, dass jedes $y$ aus der Bildmenge *höchstens einmal* abgebildet wird. +Es würden in einem Schaubild also keine zwei oder mehr Pfeile auf ein $y \in B$ treffen. +*Surjektivität* bedeutet, dass die Bildmenge die gesamte Menge $B$ umfasst, dass also $W(f) = B$ gilt. +Siehe hierzu auch die Definition von [Relationen *auf* einer Menge](../relationen/kartesisches_produkt_relationen#definitions--und-wertebereich). +*Bijektivität* schließlich bedeutet, dass jedes Element $x \in A$ auf *genau ein* Element $y \in B$ abgebildet wird und, +dass jedes $y \in B$ "belegt" ist. + +![Injekitv, surjektiv und bijektiv dargestellt](/img/mathematik/mengenlehre/relationen/bijektion.png) + +Im Schaubild ist zu erkennen, wie bei der *Injektivität* kein Element aus $B$ mehr als ein mal von einem Pfeil getroffen wird. +Es ist möglich, dass ein Element aus $B$ gar nicht getroffen, aber eben nie mehr als ein mal. +Bei *Surjektivität* ist zu sehen, wie kein Element aus $B$ frei bleibt. +Hier ist es möglich, dass mehr als ein Pfeil auf ein Element aus $B$ geht. +Es muss also mindestens ein Pfeil eingehen. +In dem Schaubild zur *Bijektivität*, als Kombination aus Beidem, ist zu sehen, dass jedes Element von $B$ genau ein mal +von einem Pfeil getroffen wird. +Es darf also kein Element aus $B$ frei bleiben und es darf nur genau ein Pfeil pro Element eingehen. + +Dass dabei $A$ und $B$ gleich viele Elemente haben müssen ist kein Zufall. +Welche Rolle bijektive Abbildungen bei der *Kardinalität* ("Größe") von Mengen haben, wird im Kapitel [Endlichkeit und Kardinalzahlen](../endlichkeit) erklärt. + +#### Beispiele +Gegeben sei $f:\ \mathbb{N} \rightarrow \mathbb{N},\ n \mapsto f(n)$. +Betrachten verschiedene Funktionsgleichungen für $f(n)$: +- $f(n)\ :=\ n+1 \quad$ (*Nachfolgerfunktion*) ist injektiv, aber nicht surjektiv, da $0 \notin W(f)$ +- $f(n)\ := \begin{cases} \lfloor \frac{n}{2} \rfloor & n \text{ gerade}\\ 0 & n \text{ ungerade} \end{cases} \quad$ ist surjektiv, + aber nicht injektiv, da z.B. $f(1) = f(3) = 0$ +- $f(n)\ :=\ n \quad$ (*identische Abbildung*) ist injektiv, surjektiv und damit auch bijektiv + +Gegeben sei $f:\ \mathbb{R} \rightarrow \mathbb{R},\ x \mapsto x^2$, also eine Parabelfunktion. +$f$ ist weder injektiv, noch surjektiv. +Wenn man allerdings die Abbildungsmengen einschränkt, kann man eine injektive, surjektive oder bijektive Abbildung erhalten. +Im Folgenden bezeichnen wir $\mathbb{R}^+$ als die Menge der nicht-negativen reellen Zahlen, also reelle Zahlen $\ge 0$. +Am einfachsten ist es, sich die folgenden Fälle zu skizzieren: +- $f:\ \mathbb{R}^+ \rightarrow \mathbb{R},\ x \mapsto x^2 \quad$ ist injektiv, aber nicht surjektiv +- $f:\ \mathbb{R} \rightarrow \mathbb{R}^+,\ x \mapsto x^2 \quad$ ist surjektiv, aber nicht injektiv +- $f:\ \mathbb{R}^+ \rightarrow \mathbb{R}^+,\ x \mapsto x^2 \quad$ ist injektiv und surjektiv und damit auch bijektiv + +Zu bijektiven Funktion sagt man auch, sie bilden "*umkehrbar eindeutig*" ab. +Eingangs wurde erwähnt: +> Jedoch ist das *Inverse* $f^{-1}$ einer Funktion $f$ im Allgemeinen keine Funktion mehr. + +Im Falle einer bijektiven Abbildung $f$ ist $f^{-1}$ allerdings eine Abbildung und man nennt diese dann *Umkehrabbildung* oder *Umkehrfunktion*. + +## Operationen +tbc \ No newline at end of file diff --git a/static/img/mathematik/mengenlehre/relationen/bijektion.png b/static/img/mathematik/mengenlehre/relationen/bijektion.png new file mode 100644 index 0000000000000000000000000000000000000000..e689d03381c1dd49a9a2080fff671f0dadac1de1 GIT binary patch literal 80487 zcmb??by(A1^gkV=81;KBAM8;-KCheRB3d 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