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

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

- the
*reflexive*property: for all in , , - the
*symmetric*property: for all in , if , then , and - the
*transitive*property: for all in , if and , then .

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* on a set is an *equivalence* iff it satisfies

- the
*reflexive*property: for all in , , and - the
*euclidean*property: for all in , if and , then .

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

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