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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: Equivalence relation
Context
Subset of
