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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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