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Es sei r := {(1, 1)}. (a) Wegen 1 r 1 gilt x r x für alle x ∈ {1}. Folglich ist r reflexiv. (b) Da 2 r 2 nicht gilt, gibt es ein x ∈ {1, 2} so, dass x r x nicht gilt. Folglich ist r nicht reflexiv. Indikatormatrix Es sei n ∈ N0 gegeben. Eine Relation r auf [1, n] ist eine Teilmenge von [1, n] × [1, n]. Die Indikatormatrix, vgl. 34), liefert eine Verbildlichung einer solchen Relation. Durch Wahl einer Abzählung, vgl. 6) Definition (Indikatormatrix). Es seien n ∈ N0 , eine Menge X, eine Abzählung e von X und eine Relation r auf X gegeben.

6) gilt folglich U− ◦ χ− = idPot(U ) . 6). 6). Insgesamt sind χ− und U− zueinander inverse Abbildungen. Beispiel. Wir haben folgende Bijektion. Pot([1, 3]) → {0, 1}3 , ∅ → (0, 0, 0), {1} → (1, 0, 0), {2} → (0, 1, 0), {3} → (0, 0, 1), {1, 2} → (1, 1, 0), {1, 3} → (1, 0, 1), {2, 3} → (0, 1, 1), {1, 2, 3} → (1, 1, 1). Endlichkeit und Kardinalität Zum Schluss dieses Abschnitts betrachten wir noch das Konzept der Kardinalität einer endlichen Menge. 37) Definition (Gleichmächtigkeit). Es seien Mengen X und Y gegeben.

Es sei eine Menge X gegeben. Die Abbildung id = idX : X → X, x → x heißt Identität (oder identische Abbildung) auf X. 15) Beispiel. Die Identität auf {1, 2, 3} ist gegeben durch id{1,2,3} : {1, 2, 3} → {1, 2, 3}, 1 → 1, 2 → 2, 3 → 3. 16) Bemerkung. Für jede Abbildung f : X → Y gilt f ◦ idX = idY ◦ f = f. Beweis. Es sei eine Abbildung f : X → Y gegeben. Dann sind f ◦ idX und idY ◦ f auch Abbildungen von X nach Y . Für alle x ∈ X gilt außerdem (f ◦ idX )(x) = f (idX (x)) = f (x), (idY ◦ f )(x) = idY (f (x)) = f (x).

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