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 equivalence_relation [2007/03/30 20:45]vkuncak equivalence_relation [2007/03/30 20:45] (current)vkuncak Both sides previous revision Previous revision 2007/03/30 20:45 vkuncak 2007/03/30 20:45 vkuncak 2007/03/30 20:44 vkuncak 2007/03/30 20:36 vkuncak 2007/03/30 20:36 vkuncak 2007/03/30 20:36 vkuncak 2007/03/30 20:34 vkuncak created 2007/03/30 20:45 vkuncak 2007/03/30 20:45 vkuncak 2007/03/30 20:44 vkuncak 2007/03/30 20:36 vkuncak 2007/03/30 20:36 vkuncak 2007/03/30 20:36 vkuncak 2007/03/30 20:34 vkuncak created Line 9: Line 9: * $x \sim y\ \land\ y \sim z\ \rightarrow\ x \sim z$ * $x \sim y\ \land\ y \sim z\ \rightarrow\ x \sim z$ - 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*} Line 20: Line 20: 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 $A_{/\sim} = P$.