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equivalence_relation [2013/05/23 17:14] nikolaj |
equivalence_relation [2014/03/21 11:11] (current) |
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===== Equivalence relation ===== | ===== Equivalence relation ===== | ||
- | ==== Definition ==== | + | ==== Set ==== |
- | | $X$ | | + | | @#55CCEE: context | @#55CCEE: $X$ | |
- | | $ \sim \in\text{Rel}(X) $ | | + | |
- | ^ $ \sim \in \text{EquivRel}(X) $ ^ | + | | @#FFBB00: definiendum | @#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 | + | | @#55CCEE: context | @#55CCEE: $ \sim \in \mathrm{Rel}(X) $ | |
+ | | $x,y,z\in X$ | | ||
- | ^ $ \forall_{\text{dom}(\sim)} x\ (\langle x,x\rangle \in \sim) $ ^ | + | | @#55EE55: postulate | @#55EE55: $ x\sim x $ | |
- | ^ $ (\langle x,y\rangle \in \sim) \Leftrightarrow (\langle y,x\rangle \in \sim) $ ^ | + | | @#55EE55: postulate | @#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: postulate | @#55EE55: $ x\sim y \land y\sim z \Leftrightarrow x\sim z $ | |
- | ==== Ramifications ==== | + | ==== Discussion ==== |
- | ==== Context ==== | + | The relation $\sim$ 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]] | ||
+ | ==== Parents ==== | ||
=== Subset of === | === Subset of === | ||
[[Reflexive relation]], [[Symmetric relation]], [[Transitive relation]] | [[Reflexive relation]], [[Symmetric relation]], [[Transitive relation]] |