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# Differences

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equivalence_relation [2007/03/30 20:45]
vkuncak
equivalence_relation [2007/03/30 20:45] (current)
vkuncak
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* $x \sim y\ \land\ y \sim z\ \rightarrow\ x \sim z$   * $x \sim y\ \land\ y \sim z\ \rightarrow\ x \sim z$

-Given an equivalence relation $\sim$, we define the set of equivalence classes $A_{\sim}$ by+Given an equivalence relation $\sim$, we define the set of equivalence classes $A_{/\sim}$ by
\begin{equation*} \begin{equation*}
-  A_{\sim} = \{ \{y \mid x \sim y\} \mid x \in A \}+  A_{/\sim} = \{ \{y \mid x \sim y\} \mid x \in A \}
\end{equation*} \end{equation*}

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x \sim y\ \iff\ \exists S \in P. \{x,y\} \subseteq S   x \sim y\ \iff\ \exists S \in P. \{x,y\} \subseteq S
\end{equation*} \end{equation*}
-is an equivalence relation such that $A_{\sim} = P$.+is an equivalence relation such that $A_{/\sim} = P$.