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equivalence_relation [2013/09/02 23:18] nikolaj |
equivalence_relation [2014/03/21 11:11] (current) |
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===== Equivalence relation ===== | ===== Equivalence relation ===== | ||
- | ==== Definition ==== | + | ==== Set ==== |
- | | @#88DDEE: $X$ | | + | | @#55CCEE: context | @#55CCEE: $X$ | |
- | | @#FFBB00: $ \sim \in \text{EquivRel}(X) $ | | + | | @#FFBB00: definiendum | @#FFBB00: $ \sim \in \text{EquivRel}(X) $ | |
+ | | @#55CCEE: context | @#55CCEE: $ \sim \in \mathrm{Rel}(X) $ | | ||
| $x,y,z\in X$ | | | $x,y,z\in X$ | | ||
- | | @#55EE55: $ \forall (u\in \text{dom}(\sim)).\ u\sim u $ | | + | | @#55EE55: postulate | @#55EE55: $ x\sim x $ | |
- | | @#55EE55: $ x\sim y \Leftrightarrow y\sim x $ | | + | | @#55EE55: postulate | @#55EE55: $ x\sim y \Leftrightarrow y\sim x $ | |
- | | @#55EE55: $ x\sim y \land y\sim z \Leftrightarrow x\sim z $ | | + | | @#55EE55: postulate | @#55EE55: $ x\sim y \land y\sim z \Leftrightarrow x\sim z $ | |
==== Discussion ==== | ==== Discussion ==== | ||
- | The relation $R$ is an equivalence relation, if it's in the intersection of all reflexive, all symmetric and all transitive relation. Hence | + | 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]] |