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Dann existiert eine Abbildung $ f: B^n \rarr B $.\\ + \ex $ f(A) = \lnot(A) $ + + \subsubsection{Prädikatenlogik} + Sei $ P $ ein Prädikat auf $ M $ + \begin{itemize} + \item $ P $ ist erfüllbar, wenn $ P(x) $ wahr ist für mindestens ein $ x \in M $ + \subitem $ \exists x \in M $ sodass $ P(x) $ wahr ist + \item $ P $ ist allgemeingültig, wenn $ P(x) $ wahr ist, wenn $ x \in M $ + \subitem $ (\forall x \in M) P(x) $ ist wahr + \end{itemize} + + \subsection{Negation von Quantoren} + \begin{itemize} + \item $ \lnot (P(x) \forall x\in M) \Lrarr \exists x \in M $ sodass $ \lnot P(x) $ + \item $ \lnot (\exists x \in M $ sodass $ P(x)) \Lrarr \forall x \in M, \lnot P(x) $ + \end{itemize} + + \subsection{Mengen und Logik - Bitvektoren} + Sei $ M := {x_1, \dots, x_n}, N \subseteq M $. Bitvektoren $ B_N := (b_1, \dots, b_n), b_j = \begin{cases} & 1, \text{ falls } x_j \in N \\ + & 0, \text{ falls } x_j \not\in N \end{cases} $ + Seien $ K $ und $ L \subseteq M, L \cap K, L \cup K $, dann sind: + \begin{itemize} + \item $ B_{L\cap K} = B_L \land B_K $ + \item $ B_{L\cup K} = B_L \lor B_K $ + \item $ B_{L^0} = 1 - B_L $ (Bitinversion) + \item $ \overline{L} = M \backslash L $ + \item $ L^0 = M \backslash L $ + \end{itemize} + \anm Die Anzahl aller möglichen Teilmengen von $ M = \{x_1, \dots, x_n\} $ (bei endlicher Potenzmenge) ist äquivalent zur Anzahl der Bitvektoren mit $ m $ Komponenten + + \subsection{Mächtigkeit} + \cdef Zwei Mengen $ M $ und $ \Omega $ heißen gleichmächtig, wenn es eine bijektive Abbildung $ f:M \rarr \Omega $ gibt\\ + \ther + \begin{enumerate}[a)] + \item Gleichmächtigkeit ist eine Äquivalenzrelation auf der Menge aller Mengen + \item Endliche Mengen mit gleicher Anzahl an Elementen sind gleichmächtig + \end{enumerate} + \ex Sei $ M := \{x_1, \dots, x_n\} $\\ + $ |P(M)| = |P(B_M)| = 2^M $ + \section{Zahlen}\label{zahlen} \subsection{Natürliche Zahlen} \subsubsection{Peano-Axiome}