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 equivalence_relation [2013/09/02 23:18]nikolaj equivalence_relation [2014/03/21 11:11] (current) 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 ​==== + ==== 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]]