From 2cb5c1cf8820d0ad01aad20bbd6808bff57be57a Mon Sep 17 00:00:00 2001 From: Niklas Date: Fri, 23 Sep 2022 17:11:38 +0200 Subject: [PATCH] Add content to 1_kartesisches_produkt_relationen.md --- assets/mathematik/mengenlehre/inverse_rel.tex | 78 +++++ .../mathematik/mengenlehre/verkettung_1.tex | 56 ++++ .../mathematik/mengenlehre/verkettung_2.tex | 89 ++++++ .../mengenlehre/relationen/1_abbildungen.md | 98 ------ .../1_kartesisches_produkt_relationen.md | 283 ++++++++++++++++++ .../mengenlehre/relationen/inverse_rel.png | Bin 0 -> 70144 bytes .../mengenlehre/relationen/verkettung_1.png | Bin 0 -> 89312 bytes .../mengenlehre/relationen/verkettung_2.png | Bin 0 -> 109603 bytes 8 files changed, 506 insertions(+), 98 deletions(-) create mode 100644 assets/mathematik/mengenlehre/inverse_rel.tex create mode 100644 assets/mathematik/mengenlehre/verkettung_1.tex create mode 100644 assets/mathematik/mengenlehre/verkettung_2.tex delete mode 100644 docs/mathematik/mengenlehre/relationen/1_abbildungen.md create mode 100644 docs/mathematik/mengenlehre/relationen/1_kartesisches_produkt_relationen.md create mode 100644 static/img/mathematik/mengenlehre/relationen/inverse_rel.png create mode 100644 static/img/mathematik/mengenlehre/relationen/verkettung_1.png create mode 100644 static/img/mathematik/mengenlehre/relationen/verkettung_2.png diff --git a/assets/mathematik/mengenlehre/inverse_rel.tex b/assets/mathematik/mengenlehre/inverse_rel.tex new file mode 100644 index 0000000..a6063f0 --- /dev/null +++ b/assets/mathematik/mengenlehre/inverse_rel.tex @@ -0,0 +1,78 @@ +\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} + + % Compose + \begin{scope} + \node at (4.5,-2.5) {\huge $\overset{R^{-1}}{\Longrightarrow}$}; + \end{scope} + + % Set names + \node at (6,-0.4) [thick] {B}; + \node at (9,-0.4) [thick] {A}; + + % Set elements + \begin{scope}[every node/.style={circle,thick,draw}] + \node (AA) at (6,-1.5) {a}; + \node (BB) at (6,-2.5) {b}; + \node (CC) at (6,-3.5) {c}; + \node (DD) at (6,-4.5) {d}; + \end{scope} + \begin{scope}[every node/.style={circle,thick,draw}] + \node (11) at (9,-1) {1}; + \node (22) at (9,-2) {2}; + \node (33) at (9,-3) {3}; + \node (44) at (9,-4) {4}; + \node (55) at (9,-5) {5}; + \end{scope} + + % Arrows + \begin{scope}[every edge/.style={draw=draculaPurple,very thick}] + \path [->] (AA) edge (11); + \path [->] (BB) edge (22); + \path [->] (BB) edge (33); + \path [->] (DD) edge (11); + \path [->] (DD) edge (44); + \end{scope} + + \end{tikzpicture} +\end{document} \ No newline at end of file diff --git a/assets/mathematik/mengenlehre/verkettung_1.tex b/assets/mathematik/mengenlehre/verkettung_1.tex new file mode 100644 index 0000000..4d00298 --- /dev/null +++ b/assets/mathematik/mengenlehre/verkettung_1.tex @@ -0,0 +1,56 @@ +\documentclass[convert={size=384}]{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} + +\begin{document} + \begin{tikzpicture}[text=draculaForeground, draw=draculaCyan] + % Set names + \node at (0,-0.4) [thick] {A}; + \node at (3,-0.4) [thick] {B}; + \node at (6,-0.4) [thick] {C}; + + % 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} + \begin{scope}[every node/.style={circle,thick,draw}] + \node (x) at (6,-2) {$\alpha$}; + \node (y) at (6,-3) {$\beta$}; + \node (z) at (6,-4) {$\gamma$}; + \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} + \begin{scope}[every edge/.style={draw=draculaPurple,very thick}] + \path [->] (A) edge (x); + \path [->] (A) edge (y); + \path [->] (A) edge (z); + \path [->] (B) edge (y); + \path [->] (C) edge (x); + \end{scope} + \end{tikzpicture} +\end{document} \ No newline at end of file diff --git a/assets/mathematik/mengenlehre/verkettung_2.tex b/assets/mathematik/mengenlehre/verkettung_2.tex new file mode 100644 index 0000000..dc91bf2 --- /dev/null +++ b/assets/mathematik/mengenlehre/verkettung_2.tex @@ -0,0 +1,89 @@ +\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}; + \node at (6,-0.4) [thick] {C}; + + % 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} + \begin{scope}[every node/.style={circle,thick,draw}] + \node (x) at (6,-2) {$\alpha$}; + \node (y) at (6,-3) {$\beta$}; + \node (z) at (6,-4) {$\gamma$}; + \end{scope} + + % Arrows + \begin{scope}[every edge/.style={draw=draculaPurple,very thick}] + \path [->] (1) edge[draw=draculaRed] (A); + \path [->] (1) edge (D); + \path [->] (2) edge[draw=draculaRed] (B); + \path [->] (3) edge[draw=draculaRed] (B); + \path [->] (4) edge (D); + \end{scope} + \begin{scope}[every edge/.style={draw=draculaPurple,very thick}] + \path [->] (A) edge[draw=draculaRed] (x); + \path [->] (A) edge[draw=draculaRed] (y); + \path [->] (A) edge[draw=draculaRed] (z); + \path [->] (B) edge[draw=draculaRed] (y); + \path [->] (C) edge (x); + \end{scope} + + % Compose + \begin{scope} + \node at (7.5,-3) {\Huge $\overset{R \circ K}{\Longrightarrow}$}; + \end{scope} + + % Set names + \node at (9,-0.4) [thick] {A}; + \node at (12,-0.4) [thick] {C}; + + % Set elements + \begin{scope}[every node/.style={circle,thick,draw}] + \node (11) at (9,-1) {1}; + \node (22) at (9,-2) {2}; + \node (33) at (9,-3) {3}; + \node (44) at (9,-4) {4}; + \node (55) at (9,-5) {5}; + \end{scope} + \begin{scope}[every node/.style={circle,thick,draw}] + \node (xx) at (12,-2) {$\alpha$}; + \node (yy) at (12,-3) {$\beta$}; + \node (zz) at (12,-4) {$\gamma$}; + \end{scope} + + % Arrows + \begin{scope}[every edge/.style={draw=draculaPurple,very thick}] + \path [->] (11) edge (xx); + \path [->] (11) edge (yy); + \path [->] (11) edge (zz); + \path [->] (22) edge (yy); + \path [->] (33) edge (yy); + \end{scope} + \end{tikzpicture} +\end{document} \ No newline at end of file diff --git a/docs/mathematik/mengenlehre/relationen/1_abbildungen.md b/docs/mathematik/mengenlehre/relationen/1_abbildungen.md deleted file mode 100644 index 3896b23..0000000 --- a/docs/mathematik/mengenlehre/relationen/1_abbildungen.md +++ /dev/null @@ -1,98 +0,0 @@ ---- -title: Kartesisches Produkt und Abbildungen -tags: [mathematik, mengenlehre, relation, kartesisches produkt] -sidebar_position: 2 ---- - -In Mengen werden Objekte zusammengefasst, allerdings ohne innere Struktur, -d.h. sie sind einfach in der Menge ohne Bedeutung und ohne Reihenfolge. -Manchmal benötigt man jedoch so etwas wie eine Reihenfolge oder -möchte Mengen miteinander in eine gewisse Verbindung bringen, ohne sie einfach nur zu Vereinigen. - -## Kartesisches Produkt -:::note Geordnete Paare - -Ein *geordnetes Paar* (*2-Tupel*) mit den Elementen $a$ und $b$ ist wie folgt als Menge definiert: -$$ - (a, b)\ :=\ \{ \overbrace{\{a, b\}}^{\text{Elemente des Tupels}}, \underbrace{\{a\}}_{\text{Erstes Element}} \} -$$ - -::: -Ein 2-Tupel fasst also je zwei Elemente zusammen und bestimmt dabei, welches zuerst kommt. -Anders als bei Mengen, bei denen $\{a,b\} = \{b,a\}$ gilt, gilt bei 2-Tupeln demnach $(a,b) \ne (b,a)$. -Die andere Schreibweise $()$, statt $\{\}$, soll dies verdeutlichen und dazu führen, dass man diese nicht verwechselt. - -Ein Verband haben wir als ein *Tripel* eingeführt. -Ein Tripel ist ein 3-Tupel. -Allgemein lassen sich rekursiv n-Tupel wie folgt definieren: - -:::note n-Tupel - -- 1-Tupel: $(x_1) = \{ x_1 \}$ -- 2-Tupel: $(x_1, x_2) = \{ \{x_1, x_2\}, \{x_1\} \}$ -- 3-Tupel: $(x_1, x_2, x_3) = ((x_1, x_2), x_3)$ -- n-Tupel: $(x_1, x_2, \dots, x_n) = ((x_1, x_2, \dots, x_{n-1}), x_n)$ - -::: -n-Tupel lassen sich also auf die Mengendarstellungen der 2-Tupel zurückführen. -In der Typentheorie nach Russell, die im Kapitel über [Mengen](../mengen) eingeführt wurde ist dies allerdings nicht erlaubt. -Denn schon bei einem 3-Tupel gibt es eine Vermischung verschiedener Mengenstufen: -$$ - \begin{align*} - (a, b, c) &= ((a, b), c)\\ - &= \{\ \{(a,b), c\},\ \{(a,b)\}\ \}\\ - &= \{\ \{\ \underbrace{\{\{a,b\}, \{a\}\}}_{(a,b)},\ c\},\ \{\underbrace{\{\{a,b\}, \{a\}\}}_{(a,b)}\}\ \}\\ - &= \{\ \{\ \underbrace{\{\{a,b\}, \{a\}\}}_{\text{Menge 2. Stufe}},\ \underbrace{c}_{\text{Menge 0. Stufe}}\},\ \{\{\{a,b\}, \{a\}\}\}\ \} - \end{align*} -$$ -Im Stufensystem ist dieser Ausdruck also nicht korrekt. -In anderen axiomatisierten Mengenlehren ist es allerdings erlaubt. -Auch für das Stufensystem gibt es korrekte Definitionen für n-Tupel. -Wir wollen es an dieser Stelle allerdings dabei belassen und nehmen es einfach hin. - -Man kann nun Tupel zu einer Menge zusammenfassen. -Die Menge aller Tupel von Mengen nennt man *kartesisches Produkt*: - -:::note Kartesisches Produkt - -Seien $A$ und $B$ Mengen, dann ist das *kartesische Produkt* der Mengen $A$ und $B$ definiert als -$$ - A \times B\ :=\ \{ (a,b)\ |\ a \in A \wedge b \in B \} -$$ - -::: - -## Abbildungen -Das kartesische Produkt erlaubt es uns jetzt Paare in einer Menge zusammenzufassen. -Daraus können wir nun eine Teilmenge betrachten. - -:::note Relation - -Seien $A$, $B$ Mengen. -Man nennt $R$ *Relation* oder *Korrespondenz* __aus__ $A$ __in__ $B$ $:=\ R \subseteq A \times B$ - -::: - -#### Beispiele -Nehmen wir eine Menge mit drei Elementen $A = \{a_1, a_2, a_3\}$ und bilden alle möglichen Paare. -Diese Paare fassen wir nun in einer Menge zusammen: -$$ - A \times A = \{(a_1, a_1), (a_1, a_2), (a_1, a_3),\quad (a_2, a_1), (a_2, a_2), (a_2, a_3),\quad (a_3, a_1), (a_3, a_2), (a_3, a_3)\} -$$ -Daraus können wir jetzt bspw. unterschiedliche Relationen basteln: -$$ - \begin{align*} - R_0 &= \emptyset\\ - R_1 &= \{(a_1, a_1), (a_1, a_3)\}\\ - R_2 &= \{(a_1, a_1), (a_1, a_3), (a_2, a_2), (a_2, a_3), (a_3, a_3)\}\\ - R_3 &= \{(a_2, a_1), (a_2, a_2), (a_2, a_3)\}\\ - R_4 &= A \times A - \end{align*} -$$ - -Gegeben sei $A = \{1, 2, 3, 4, 5\}$, $B = \{a, b, c, d\}$ und $R = \{(1, a), (1,d), (2, b), (3, b), (5, d)\} \subseteq A \times B$. -Man kann das veranschaulichen, wie die jeweiligen Elemente in Beziehung stehen: - -![Relation R veranschaulicht durch Pfeile](/img/mathematik/mengenlehre/relationen/relation.png) - -*tbc* \ No newline at end of file diff --git a/docs/mathematik/mengenlehre/relationen/1_kartesisches_produkt_relationen.md b/docs/mathematik/mengenlehre/relationen/1_kartesisches_produkt_relationen.md new file mode 100644 index 0000000..3663647 --- /dev/null +++ b/docs/mathematik/mengenlehre/relationen/1_kartesisches_produkt_relationen.md @@ -0,0 +1,283 @@ +--- +title: Kartesisches Produkt und Relationen +tags: [mathematik, mengenlehre, relation, kartesisches produkt] +sidebar_position: 2 +--- + +In Mengen werden Objekte zusammengefasst, allerdings ohne innere Struktur, +d.h. sie sind einfach in der Menge ohne Bedeutung und ohne Reihenfolge. +Manchmal benötigt man jedoch so etwas wie eine Reihenfolge oder +möchte Mengen miteinander in eine gewisse Verbindung bringen, ohne sie einfach nur zu Vereinigen. + +## Kartesisches Produkt +:::note Geordnete Paare + +Ein *geordnetes Paar* (*2-Tupel*) mit den Elementen $a$ und $b$ ist wie folgt als Menge definiert: +$$ + (a, b)\ :=\ \{ \overbrace{\{a, b\}}^{\text{Elemente des Tupels}}, \underbrace{\{a\}}_{\text{Erstes Element}} \} +$$ + +::: +Ein 2-Tupel fasst also je zwei Elemente zusammen und bestimmt dabei, welches zuerst kommt. +Anders als bei Mengen, bei denen $\{a,b\} = \{b,a\}$ gilt, gilt bei 2-Tupeln demnach $(a,b) \ne (b,a)$. +Die andere Schreibweise $()$, statt $\{\}$, soll dies verdeutlichen und dazu führen, dass man diese nicht verwechselt. + +Ein Verband haben wir als ein *Tripel* eingeführt. +Ein Tripel ist ein 3-Tupel. +Allgemein lassen sich rekursiv n-Tupel wie folgt definieren: + +:::note n-Tupel + +- 1-Tupel: $(x_1) = \{ x_1 \}$ +- 2-Tupel: $(x_1, x_2) = \{ \{x_1, x_2\}, \{x_1\} \}$ +- 3-Tupel: $(x_1, x_2, x_3) = ((x_1, x_2), x_3)$ +- n-Tupel: $(x_1, x_2, \dots, x_n) = ((x_1, x_2, \dots, x_{n-1}), x_n)$ + +::: +n-Tupel lassen sich also auf die Mengendarstellungen der 2-Tupel zurückführen. +In der Typentheorie nach Russell, die im Kapitel über [Mengen](../mengen) eingeführt wurde ist dies allerdings nicht erlaubt. +Denn schon bei einem 3-Tupel gibt es eine Vermischung verschiedener Mengenstufen: +$$ + \begin{align*} + (a, b, c) &= ((a, b), c)\\ + &= \{\ \{(a,b), c\},\ \{(a,b)\}\ \}\\ + &= \{\ \{\ \underbrace{\{\{a,b\}, \{a\}\}}_{(a,b)},\ c\},\ \{\underbrace{\{\{a,b\}, \{a\}\}}_{(a,b)}\}\ \}\\ + &= \{\ \{\ \underbrace{\{\{a,b\}, \{a\}\}}_{\text{Menge 2. Stufe}},\ \underbrace{c}_{\text{Menge 0. Stufe}}\},\ \{\{\{a,b\}, \{a\}\}\}\ \} + \end{align*} +$$ +Im Stufensystem ist dieser Ausdruck also nicht korrekt. +In anderen axiomatisierten Mengenlehren ist es allerdings erlaubt. +Auch für das Stufensystem gibt es korrekte Definitionen für n-Tupel. +Wir wollen es an dieser Stelle allerdings dabei belassen und nehmen es einfach hin. + +Man kann nun Tupel zu einer Menge zusammenfassen. +Die Menge aller Tupel von Mengen nennt man *kartesisches Produkt*: + +:::note Kartesisches Produkt + +Seien $A$ und $B$ Mengen, dann ist das *kartesische Produkt* der Mengen $A$ und $B$ definiert als +$$ + A \times B\ :=\ \{ (a,b)\ |\ a \in A \wedge b \in B \} +$$ + +::: + +## Relationen +Das kartesische Produkt erlaubt es uns jetzt Paare in einer Menge zusammenzufassen. +Daraus können wir nun eine Teilmenge betrachten. + +:::note Relation + +Seien $A$, $B$ Mengen. +Man nennt $R$ *Relation* oder *Korrespondenz* __aus__ $A$ __in__ $B$ $:=\ R \subseteq A \times B$ + +::: + +#### Beispiele +Nehmen wir eine Menge mit drei Elementen $A = \{a_1, a_2, a_3\}$ und bilden alle möglichen Paare. +Diese Paare fassen wir nun in einer Menge zusammen: +$$ + A \times A = \{(a_1, a_1), (a_1, a_2), (a_1, a_3),\quad (a_2, a_1), (a_2, a_2), (a_2, a_3),\quad (a_3, a_1), (a_3, a_2), (a_3, a_3)\} +$$ +Daraus können wir jetzt bspw. unterschiedliche Relationen basteln: +$$ + \begin{align*} + R_0 &= \emptyset\\ + R_1 &= \{(a_1, a_1), (a_1, a_3)\}\\ + R_2 &= \{(a_1, a_1), (a_1, a_3), (a_2, a_2), (a_2, a_3), (a_3, a_3)\}\\ + R_3 &= \{(a_2, a_1), (a_2, a_2), (a_2, a_3)\}\\ + R_4 &= A \times A + \end{align*} +$$ + +Gegeben sei $A = \{1, 2, 3, 4, 5\}$, $B = \{a, b, c, d\}$ und $R = \{(1,a), (1,d), (2,b), (3,b), (4,d)\} \subseteq A \times B$. +Man kann auch veranschaulichen, wie die jeweiligen Elemente in Beziehung stehen: + +![Relation R veranschaulicht durch Pfeile](/img/mathematik/mengenlehre/relationen/relation.png) + +Einige Begriffe die jetzt noch kommen sind vielleicht schon aus der Schulmathematik bekannt. +Dort hat man sie allerdings für *Funktionen* kennengelernt. +Da Funktionen spezielle Relationen sind, ist das nicht weiter schlimm. +Dennoch sollte man darauf achten, ob man gerade mit einer echten Funktion oder nur einer Relation hantiert, +wenn man mit diesen Begriffen jongliert. +Funktionen werden aber sehr bald schon eingeführt. + +### Definitions- und Wertebereich +Es ist sinnvoll Mengen zu definieren, die nur die Elemente ersten oder die zweiten Elemente einer Relation erfasst: + +:::note Definitions- und Wertebereich + +- *Definitionsbereich* von $R$: + $$ + D(R)\ :=\ \{ x\ |\ x \in A \wedge \exists y \in B:\ (x,y) \in R \} + $$ + Der Definitionsbereich von $R$ umfasst also alle Elemente $x$ aus $A$, für die es ein $y$ aus $B$ gibt. +- *Wertebereich* oder *Bild* von $R$: + $$ + W(R)\ :=\ \{ y\ |\ y \in B \wedge \exists x \in A:\ (x,y) \in R \} + $$ + Der Wertebereich oder das Bild von $R$ umfasst also alle Elemente $y$ aus $B$, für die es ein $x$ aus $A$ gibt. + +::: + +#### Beispiele +Gegeben sei $A = \{1, 2, 3, 4, 5\}$, $B = \{a, b, c, d\}$ und $R = \{(1,a), (1,d), (2,b), (3,b), (4,d)\} \subseteq A \times B$. +Dann sind: +- $D(R) = \{ 1, 2, 3, 5 \}$ +- $W(R) = \{ a, b, d \}$ + +Die Elemente $5 \in A$ und $c \in B$ tauchen nicht auf, da für diese jeweils kein $x$ oder $y$ existiert, +sie kommen also nirgends in $R$ vor. + +Gegeben sei $A = \{a_1, a_2, a_3\}$ und folgende Relationen $R_i \subseteq A \times A$ für $i=0,\dots,4$: +$$ + \begin{align*} + R_0 &= \emptyset\\ + R_1 &= \{(a_1, a_1), (a_1, a_3)\}\\ + R_2 &= \{(a_1, a_1), (a_1, a_3), (a_2, a_2), (a_2, a_3), (a_3, a_3)\}\\ + R_3 &= \{(a_2, a_1), (a_2, a_2), (a_2, a_3)\}\\ + R_4 &= A \times A + \end{align*} +$$ +Dann sind: +- $D(R_0) = \emptyset$ und $W(R_0) = \emptyset$ +- $D(R_1) = \{ a_1 \}$ und $W(R_1) = \{ a_1, a_3\}$ +- $D(R_2) = \{ a_1, a_2, a_3 \}$ und $W(R_2) = \{ a_1, a_2, a_3 \}$ +- $D(R_3) = \{ a_2 \}$ und $W(R_3) = \{ a_1, a_2, a_3 \}$ +- $D(R_4) = W(R_4) = A$ + +Je nachdem, ob der Definitions- oder Wertebereich die gesamte Menge $A$ bzw. $B$ umfasst gibt es verschiedene Sprechweisen. + +:::note Sprechweisen + +$R$ ist Relation +- "*__von__* $A$ *in* $B$" $:=\ D(R) = A$ +- "*aus* $A$ *__auf__* $B$" $:=\ W(R) = B$ +- "*__von__* $A$ *__auf__* $B$" $:=\ D(R) = A \wedge W(R) = B$ + +::: + +Wenn nun eine Relation gegeben ist, dann möchte man vielleicht die Frage stellen "Welche $y$-Werte werden für ein $x$ angenommen?" +und möchte dies formal ausdrücken bzw. aufschreiben, ohne dabei immer die Tupel angeben zu müssen. + +:::note + +Sei $R$ Relation. +$$ + R(x)\ :=\ \{ y\ |\ y \in B \wedge (x,y) \in R \} +$$ + +::: + +#### Beispiele +Gegeben sei $A = \{1, 2, 3, 4, 5\}$, $B = \{a, b, c, d\}$ und $R = \{(1,a), (1,d), (2,b), (3,b), (4,d)\} \subseteq A \times B$. +Dann ist: +$$ + \begin{align*} + R(1) &= \{ a, d \}\\ + R(2) &= \{ b \}\\ + R(3) &= \{ b \}\\ + R(4) &= \{ d \}\\ + R(5) &= \emptyset + \end{align*} +$$ + +### Verkettung und Inverse +Man kann auch zwei Relationen kombinieren, sofern diese *in* und *aus* den gleichen Mengen korrespondieren - in dieser Reihenfolge. + +:::note Verkettung / Relationenprodukt + +Seien $R \subseteq A \times B$ und $K \subseteq B \times C$ Relationen. +Dann ist die Verkettung von $R$ mit $K$ definiert als: +$$ + F \circ K\ :=\ \{ (x,z)\ |\ \exists y \in B:\ (x,y) \in R \wedge (y,z) \in K \} +$$ +Also ist $F \circ K \subseteq A \times C$. + +::: + +#### Beispiele +Gegeben seien $A = \{1, 2, 3, 4, 5\}$, $B = \{a, b, c, d\}$ und $C = \{\alpha, \beta, \gamma\}$. +$R = \{(1,a), (1,d), (2,b), (3,b), (4,d)\} \subseteq A \times B$ und +$K = \{(a, \alpha), (a, \beta), (a, \gamma), (b, \beta), (c, \alpha)\} \subseteq B \times C$ seien Relationen. +Als Bild veranschaulicht: + +![Relation R und K veranschaulicht durch Pfeile](/img/mathematik/mengenlehre/relationen/verkettung_1.png) + +Die Verkettung sind dann die Paare, die gebildet werden können, wenn man alle möglichen Pfeile in den schaubildern entlang geht. +$$ + F \circ K = \{(1, \alpha), (1, \beta), (1, \gamma), (2, \beta), (3, \beta)\} +$$ +Im folgenden Schaubild sind die einzelnen Relationen aufgeführt, nur dieses sind die Pfeile rot eingefärbt, +die für die Verkettung relevant sind. +Jeden dieser Pfeile geht man entlang und sammelt die Paare. +Rechts im Bild ist dann nur noch das Resultat der Verkettung $R \circ K$ zu sehen: + +![Verkettung R mit K veranschaulicht durch Pfeile](/img/mathematik/mengenlehre/relationen/verkettung_2.png) + +Relationen kann man auch umkehren. +Die Rolle der ersten Elemente eines Tupels vertauscht sich dann mit den zweiten Elementen. +Das heißt, Wenn $x$ mit $y$ in Relation steht, also $(x,y) \in R$, dann ist das *Inverse* dazu $(y,x) \in R^{-1}$. + +:::note Inverse Relation + +Gegeben sei $R \subseteq A \times B$ +$$ + R^{-1}\ :=\ \{(y,x)\ |\ (x,y) \in R \} \subset B \times A +$$ +ist die *inverse Relation* zu $R$. + +::: +Man beachte, dass sich entsprechend das kartesische Produkt umdreht. +Also wenn $R \subseteq A \times B$, dann ist $R^{-1} \subseteq B \times A$. + +#### Beispiele +Gegeben seien $A = \{1, 2, 3, 4, 5\}$, $B = \{a, b, c, d\}$ und $R = \{(1,a), (1,d), (2,b), (3,b), (4,d)\} \subseteq A \times B$. +Das Inverse $R^{-1}$ von $R$ ist jetzt jedes Paar aus $R$ umgedreht: +$$ + R^{-1} = \{ a,1), (b,2), (b,3), (d,1), (d,4) \} +$$ +Veranschaulicht: + +![Relation R und Inverse von R veranschaulicht durch Pfeile](/img/mathematik/mengenlehre/relationen/inverse_rel.png) + +Für die Verkettung und das Inverse gelten nun folgende Eigenschaften: +:::tip Sätze über Verkettung und Inverse von Relationen + +1. Die Verkettung ist assoziativ: $(R \circ K) \circ F = R \circ (K \circ F )$ +2. $(R^{-1})^{-1} = R$ +3. $(R \circ K)^{-1} = K^{-1} \circ R^{-1}$ + +::: +Die Assoziativität ist leicht zu beweisen. +Man geht auf die Definitionsebene mit Mengen und definierenden Ausdrücken und führt es auf die Assoziativität der logischen Operatoren zurück, +sowie es bereits bei der [Schnittmenge der Mengenalgebra](../mengen#beweis) gemacht wurde. + +Der Beweis für den zweiten Satz ist *trivial*. +Trivial ist eine oft benutzte Bezeichnung in der Mathematik, wenn etwas offensichtlich oder leicht einsehbar ist, +also so leicht, dass man keinen extra beweis dafür angibt - dennoch gibt es Fallen und es sollte nicht leichtfertig benutzt werden. +In diesem Fall mache man sich klar, dass 2. wirklich gilt und beweise es sogar. + +#### Beweis von 3 +Der dritte Satz ist eine schöne Beziehung zwischen Verkettung und den inversen Relationen. +$$ +\begin{alignat*}{2} + \quad && &(z,x) \in (R \circ K)^{-1}\\ + \overset{\text{Def. } \square^{-1}}{\Longleftrightarrow}\quad && &(x,z) \in R \circ K\\ + \overset{\text{Def. } \circ}{\Longleftrightarrow}\quad && &\exists y:\ (x,y) \in R \wedge (y,z) \in K\\ + \overset{\text{Def. } \square^{-1}}{\Longleftrightarrow}\quad && &\exists y:\ (y,x) \in R^{-1} \wedge (z,y) \in K^{-1}\\ + \overset{\text{Kom. } \wedge}{\Longleftrightarrow}\quad && &\exists y:\ (z,y) \in K^{-1} \wedge (y,x) \in R^{-1}\\ + \overset{\text{Def. } \circ}{\Longleftrightarrow}\quad && &(z,x) \in K^{-1} \circ R^{-1} + \qquad\qquad\qquad\qquad\blacksquare +\end{alignat*} +$$ +Die $\Longleftrightarrow$ sind symbolische Schreibweisen für "... genau dann, wenn ..." und beziehen sich auf die Logik. +Statt $\longleftrightarrow$, wurde hier $\Longleftrightarrow$ genutzt. +Beweise in der Mathematik sind eher Skizzen und kein formaler Beweis, wie man ihn in der Logik korrekt führen müsste +und manchmal verschwimmen dabei auch die Objekt- und Metasprache. + +Jeder Schritt der hier gemacht 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