Differences
This shows you the differences between two versions of the page.
| Both sides previous revision Previous revision Next revision | Previous revision Next revision Both sides next revision | ||
|
equivalence_relation [2013/05/23 17:14] nikolaj |
equivalence_relation [2013/09/02 23:18] nikolaj |
||
|---|---|---|---|
| Line 1: | Line 1: | ||
| ===== Equivalence relation ===== | ===== Equivalence relation ===== | ||
| ==== Definition ==== | ==== Definition ==== | ||
| - | | $X$ | | + | | @#88DDEE: $X$ | |
| - | | $ \sim\ \in\text{Rel}(X) $ | | + | |
| - | ^ $ \sim\ \in \text{EquivRel}(X) $ ^ | + | | @#FFBB00: $ \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 | + | | $x,y,z\in X$ | |
| - | ^ $ \forall_{\text{dom}(\sim)} x\ (\langle x,x\rangle \in\ \sim) $ ^ | + | | @#55EE55: $ \forall (u\in \text{dom}(\sim)).\ u\sim u $ | |
| - | ^ $ (\langle x,y\rangle \in\ \sim) \Leftrightarrow (\langle y,x\rangle \in\ \sim) $ ^ | + | | @#55EE55: $ x\sim y \Leftrightarrow y\sim x $ | |
| - | ^ $ (\langle x,y\rangle \in\ \sim) \land (\langle y,z\rangle \in\ \sim) \Leftrightarrow (\langle x,z\rangle \in\ \sim) $ ^ | + | | @#55EE55: $ x\sim y \land y\sim z \Leftrightarrow x\sim z $ | |
| - | ==== Ramifications ==== | + | ==== Discussion ==== |
| + | The relation $R$ is an equivalence relation, if it's in the intersection of all reflexive, all symmetric and all transitive relation. Hence | ||
| + | === Reference === | ||
| + | Wikipedia: [[http://en.wikipedia.org/wiki/Equivalence_relation|Equivalence relation]] | ||
| ==== Context ==== | ==== Context ==== | ||
| === Subset of === | === Subset of === | ||
| [[Reflexive relation]], [[Symmetric relation]], [[Transitive relation]] | [[Reflexive relation]], [[Symmetric relation]], [[Transitive relation]] | ||
