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A relation is an equivalence iff it is reflexive and euclidean

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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