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

Question

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

Answer 1

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 $S$.

An equivalence class is a collection of elements which are all related to each other under an equivalence relation. It follows from the definition of an equivalence relation that sets can be partitioned into pairwise disjoint equivalence classes given any equivalence relation on that set.

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 $\mathbb{Z}=[0]\cup [1]$.