From c188e082751a279577ce84702744618387721225 Mon Sep 17 00:00:00 2001
From: Gregor Kleen <gkleen@yggdrasil.li>
Date: Fri, 22 Apr 2016 23:19:04 +0200
Subject: lds 01

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 ss2016/lds/01/H1-3.md | 33 +++++++++++++++++++++++++++++++++
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diff --git a/ss2016/lds/01/H1-3.md b/ss2016/lds/01/H1-3.md
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+a) Disjunkte Mengen bilden ein Gegenbeispiel:
+
+\begin{align*}
+A &= \{ a \} \\
+B &= \{ b \} \\
+C &= \{ c \} \\
+D &= \{ d \} \\
+A - (B - (C - D)) &= \{ a \} \\
+(A \cup C) - (B \cup D) &= \{ a, c \}
+\end{align*}
+
+<!-- Es sei $\Omega = A \cup B \cup C \cup D$ -->
+
+<!-- \begin{align*} -->
+<!-- \ & A - (B - (C - D)) = (A \cup C) - (B \cup D) \\ -->
+<!-- \underset{\text{Extens.}}{\lequiv}& \forall x \in \Omega .\ \iota_A(x) \land \lnot (\iota_B(x) \land \lnot (\iota_C(x) \land \lnot \iota_D(x))) \lequiv (\iota_A(x) \lor \iota_C(x)) \land \lnot (\iota_B(x) \lor \iota_D(x)) \\ -->
+<!-- \lequiv& \forall x \in \Omega .\ \iota_A(x) \land (\lnot \iota_B(x) \lor (\iota_C(x) \land \lnot \iota_D(x))) \lequiv (\iota_A(x) \lor \iota_C(x)) \land (\lnot \iota_B(x) \land \lnot \iota_D(x)) \\ -->
+<!-- \lequiv& \forall x \in \Omega .\ (\iota_A(x) \land \lnot \iota_B(x)) \lor (\iota_A(x) \land \iota_C(x) \land \lnot \iota_D(x))) \lequiv (\iota_A(x) \land \lnot \iota_B(x) \land \lnot \iota_D(x)) \lor (\lnot \iota_B(x) \land \iota_C(x) \land \lnot \iota_D(x)) -->
+<!-- \end{align*} -->
+
+b) Es sei $\Omega = A \cup B$ und für eine Menge $X$ sei $\bar X = \Omega - X$.
+$\Omega$ bildet mit $\cap, \cup$ und dem soeben definierten Komplement eine boolsche Algebra.
+
+Die Differenz $A - B$ ist definiert als $A \cap \bar B$.
+
+\begin{align*}
+(A - B) \cup (B - A) &\underset{\text{\scriptsize Diff.}}{=} (A \cap \bar B) \cup (B \cap \bar A) \\
+&\underset{\text{\scriptsize Dist.}}{=} ((A \cap \bar B) \cup B) \cap ((A \cap \bar B) \cup \bar A) \\
+&\underset{\text{\scriptsize Dist.}}{=} ((A \cup B) \cap (\bar B \cup B)) \cap ((A \cup \bar A) \cap (\bar B \cup \bar A)) \\
+&\underset{\text{\scriptsize Tnd.\footnotemark}}{=} (A \cup B) \cap (\bar A \cup \bar B) \\
+&\underset{\text{\scriptsize Diff.}}{=} (A \cup B) - (A \cap B)
+\end{align*}
+\footnotetext{Tertium non datur}
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