From dbb71a8d185eb9ad078163b1203d759f946b3d26 Mon Sep 17 00:00:00 2001 From: Niklas Date: Thu, 22 Sep 2022 18:52:39 +0200 Subject: [PATCH] Add content to 1_kartesisches_produkt.md --- .../relationen/1_kartesisches_produkt.md | 93 +++++++++++++++++- .../mengenlehre/relationen/relation.png | Bin 0 -> 49498 bytes 2 files changed, 92 insertions(+), 1 deletion(-) create mode 100644 static/img/mathematik/mengenlehre/relationen/relation.png diff --git a/docs/mathematik/mengenlehre/relationen/1_kartesisches_produkt.md b/docs/mathematik/mengenlehre/relationen/1_kartesisches_produkt.md index 81508cb..3896b23 100644 --- a/docs/mathematik/mengenlehre/relationen/1_kartesisches_produkt.md +++ b/docs/mathematik/mengenlehre/relationen/1_kartesisches_produkt.md @@ -4,4 +4,95 @@ tags: [mathematik, mengenlehre, relation, kartesisches produkt] sidebar_position: 2 --- -WIP \ No newline at end of file +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 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