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

---
 ss2016/lds/01/H1-1.md  | 24 ++++++++++++++++++++++++
 ss2016/lds/01/H1-2.md  | 10 ++++++++++
 ss2016/lds/01/H1-3.md  | 33 +++++++++++++++++++++++++++++++++
 ss2016/lds/01/manifest |  3 +++
 4 files changed, 70 insertions(+)
 create mode 100644 ss2016/lds/01/H1-1.md
 create mode 100644 ss2016/lds/01/H1-2.md
 create mode 100644 ss2016/lds/01/H1-3.md
 create mode 100644 ss2016/lds/01/manifest

(limited to 'ss2016/lds')

diff --git a/ss2016/lds/01/H1-1.md b/ss2016/lds/01/H1-1.md
new file mode 100644
index 0000000..e84b407
--- /dev/null
+++ b/ss2016/lds/01/H1-1.md
@@ -0,0 +1,24 @@
+\begin{center}
+\begin{tikzpicture}[level 1/.style={level distance=1.5cm}]
+\Tree
+[.$\lequiv$ 
+  [.$\limplies$
+    [.$A$ ]
+    [.$B$ ]
+  ]
+  [.$\limplies$
+    [.$C$ ]
+    [.$\limplies$
+      [.$\lnot$ $D$ ]
+      [.$\lor$
+        [.$E$ ]
+        [.$\land$
+          [.$F$ ]
+          [.$G$ ]
+        ]
+      ]
+    ]
+  ]
+]
+\end{tikzpicture}
+\end{center}
diff --git a/ss2016/lds/01/H1-2.md b/ss2016/lds/01/H1-2.md
new file mode 100644
index 0000000..43a35d1
--- /dev/null
+++ b/ss2016/lds/01/H1-2.md
@@ -0,0 +1,10 @@
+\begin{prooftree}
+\AxiomC{$A \seq A$}
+\RightLabel{\scriptsize $\lnot\ \text{R}$}
+\UnaryInfC{$\seq A, \lnot A$}
+\AxiomC{$A \seq A$}
+\RightLabel{\scriptsize $\limplies\ \text{L}$}
+\BinaryInfC{\lnot A \limplies A \seq A}
+\RightLabel{\scriptsize $\limplies\ \text{R}$}
+\UnaryInfC{$(\lnot A \limplies A) \seq A$}
+\end{prooftree}
diff --git a/ss2016/lds/01/H1-3.md b/ss2016/lds/01/H1-3.md
new file mode 100644
index 0000000..b610a02
--- /dev/null
+++ b/ss2016/lds/01/H1-3.md
@@ -0,0 +1,33 @@
+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}
diff --git a/ss2016/lds/01/manifest b/ss2016/lds/01/manifest
new file mode 100644
index 0000000..f6428d6
--- /dev/null
+++ b/ss2016/lds/01/manifest
@@ -0,0 +1,3 @@
+H1-1.pdf
+H1-2.pdf
+H1-3.pdf
\ No newline at end of file
-- 
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