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 equivalence_relation [2013/05/23 17:19]nikolaj equivalence_relation [2013/09/04 17:27]nikolaj Both sides previous revision Previous revision 2013/09/04 17:27 nikolaj 2013/09/02 23:18 nikolaj 2013/05/23 17:19 nikolaj 2013/05/23 17:14 nikolaj 2013/05/23 17:14 nikolaj 2013/05/23 17:14 nikolaj 2013/05/23 17:13 nikolaj 2013/05/23 17:13 nikolaj 2013/05/23 17:05 nikolaj 2013/05/23 17:04 nikolaj created Next revision Previous revision 2013/09/04 17:27 nikolaj 2013/09/02 23:18 nikolaj 2013/05/23 17:19 nikolaj 2013/05/23 17:14 nikolaj 2013/05/23 17:14 nikolaj 2013/05/23 17:14 nikolaj 2013/05/23 17:13 nikolaj 2013/05/23 17:13 nikolaj 2013/05/23 17:05 nikolaj 2013/05/23 17:04 nikolaj created 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 + | @#​88DDEE: ​$\sim \in \mathrm{Rel}(X) ​$ | + | $x,y,z\in X$ | - ^ $\forall_{\text{dom}(\sim)} x\ (\langle x,x\rangle \in \sim)$ ^ + | @#​55EE55: ​$x\sim x$ | - ^ $(\langle ​x,y\rangle \in \sim) \Leftrightarrow ​(\langle ​y,x\rangle \in \sim)$ ^ + | @#​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 $\sim$ 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]]