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equivalence_relation [2013/05/23 17:19] 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 ==== |
| + | The relation $\sim$ is an equivalence relation, if it's in the intersection of all reflexive, all symmetric and all transitive relation. Hence | ||
| === Reference === | === Reference === | ||
| Wikipedia: [[http://en.wikipedia.org/wiki/Equivalence_relation|Equivalence relation]] | Wikipedia: [[http://en.wikipedia.org/wiki/Equivalence_relation|Equivalence relation]] | ||
| - | ==== Context ==== | + | ==== Parents ==== |
| === Subset of === | === Subset of === | ||
| [[Reflexive relation]], [[Symmetric relation]], [[Transitive relation]] | [[Reflexive relation]], [[Symmetric relation]], [[Transitive relation]] | ||
