# What is the difference between a binary relation and an equivalence class?

Is an equivalence class essentially a binary relation whose elements have an equivalence relation?

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have you gone to wikipedia? or, even done a google search for the terms? What is an 'equivalence relation'? – Erik G. Jan 18 '13 at 1:19
– Erik G. Jan 18 '13 at 1:22

The difference between an equivalence relation and a general binary relation is simply in the definition: equivalence relations have to behave like one would expect an $=$ sign would. $a$ always equals $a$, $a$ equals $b$ anytime $b$ equals $a$, and $a$ equals $c$ whenever $a$ equals $b$ and $b$ equals $c$. Thus if a binary relation $\sim$ on a set $S$ satisfies $a\sim a$ for all $a\in S$, $a\sim b \Rightarrow b \sim a$, and $(a\sim b )\wedge(b\sim c) \Rightarrow a \sim c$, we say that $\sim$ is an equivalence relation on $S$.
So, for example, under the relation $a\sim b$ if $a\equiv b \pmod{2}$, the set of integers has two equivalence classes - the even numbers and the odd numbers. You write equivalence classes by picking some member to represent the set, then writing that member in brackets, like $[x]$. So under the above equivalence relation $\mathbb{Z}=[0]\cup [1]$.