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High-school students and undergraduates are (almost) always taught the following definition of an equivalence relation.

A binary relation R on a set A is an equivalence iff it satisfies

  • the reflexive property: for all a  in A, a R a,
  • the symmetric property: for all a, b in A, if a R b, then b R a, and
  • the transitive property: for all a, b, c in A, if a R b and b R c, then a R c.

However, there is another formulation of an equivalence relation that one usually doesn’t hear about, as far as I know. And, it is the following one.

A binary relation R on a set A is an equivalence iff it satisfies

  • the reflexive property: for all a  in A, a R a, and
  • the euclidean property: for all a, b, c in A, if a R b and a R c, then b R c.

Exercise:  Show that a binary relation R on a set A is reflexive, symmetric and transitive iff it is reflexive and euclidean.


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