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equivalence_relation [2013/05/23 17:14] nikolaj |
equivalence_relation [2013/05/23 17:14] nikolaj |
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| ==== Definition ==== | ==== Definition ==== | ||
| | $X$ | | | $X$ | | ||
| - | | $ \sim\ \in\text{Rel}(X) $ | | + | | $ \sim \in\text{Rel}(X) $ | |
| - | ^ $ \sim\ \in \text{EquivRel}(X) $ ^ | + | ^ $ \sim \in \text{EquivRel}(X) $ ^ |
| The relation $R$ is an equivalence relation, if it's in the intersection of all reflexive, all symmetric and all transitive relation. Hence | The relation $R$ is an equivalence relation, if it's in the intersection of all reflexive, all symmetric and all transitive relation. Hence | ||
| - | ^ $ \forall_{\text{dom}(\sim)} x\ (\langle x,x\rangle \in\ \sim) $ ^ | + | ^ $ \forall_{\text{dom}(\sim)} x\ (\langle x,x\rangle \in \sim) $ ^ |
| - | ^ $ (\langle x,y\rangle \in\ \sim) \Leftrightarrow (\langle y,x\rangle \in\ \sim) $ ^ | + | ^ $ (\langle x,y\rangle \in \sim) \Leftrightarrow (\langle y,x\rangle \in \sim) $ ^ |
| - | ^ $ (\langle x,y\rangle \in\ \sim) \land (\langle y,z\rangle \in\ \sim) \Leftrightarrow (\langle x,z\rangle \in\ \sim) $ ^ | + | ^ $ (\langle x,y\rangle \in \sim) \land (\langle y,z\rangle \in \sim) \Leftrightarrow (\langle x,z\rangle \in \sim) $ ^ |
| ==== Ramifications ==== | ==== Ramifications ==== | ||
