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$A\times B$ ist eine Teilmenge des kartesischen Produkts. + + $R \subseteq A\times B$ + +\subsection{Äquivalenzrelation} + Eine Äquivalenzrelation ist eine zweistellige Relation auf einer Ausgangsmenge M mit folgenden drei Eigenschaften: + +\begin{itemize} + +\item \textbf{Reflexivität} + + Jedes Element der Ausgangsmenge M steht sich selbst in Beziehung. + + \begin{quote} + Für alle $a \in M$ gilt $\left( a , a \right) \in R$ + \end{quote} + +\item \textbf{Symmetrie} + + Zu jedem Paar $\left( a , b \right)$ ist auch die Umkehrung in $R$ enthalten. + + \begin{quote} + Wenn $\left( a , b \right) \in R$, dann ist auch $\left( b , a \right) \in R$ + \end{quote} + +\item \textbf{Transitivität} + + Stehen drei Elemente verkettet in Beziehung, dann stehen sie auch direkt in Beziehung. + + \begin{quote} + Wenn $\left( a , b \right) , \left( b , c \right) \in R$ dann ist auch $\left(a , c \right) \in R$ + \end{quote} + +\end{itemize} + \end{sectionbox} % Aussagenlogik