Differences
This shows you the differences between two versions of the page.
|
equivalence_relation [2013/05/23 17:19] nikolaj |
equivalence_relation [2014/03/21 11:11] |
||
|---|---|---|---|
| Line 1: | Line 1: | ||
| - | ===== Equivalence relation ===== | ||
| - | ==== Definition ==== | ||
| - | | $X$ | | ||
| - | | $ \sim \in\text{Rel}(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 | ||
| - | |||
| - | ^ $ \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) \land (\langle y,z\rangle \in \sim) \Leftrightarrow (\langle x,z\rangle \in \sim) $ ^ | ||
| - | |||
| - | ==== Ramifications ==== | ||
| - | === Reference === | ||
| - | Wikipedia: [[http://en.wikipedia.org/wiki/Equivalence_relation|Equivalence relation]] | ||
| - | ==== Context ==== | ||
| - | === Subset of === | ||
| - | [[Reflexive relation]], [[Symmetric relation]], [[Transitive relation]] | ||
