Equivalence Class of the relation {(0,0) (1,1) (2,2) (3,3)}

Question

The above relation is equivalent for the set {0,1,2,3}. How would you find the equivalence class for this relation or any general relational set of pairs of integers?

Answer 1

Check that each element of $\,\{0, 1,2,3\}\,$ belongs to exactly one singleton subset of the given relation and, thus, this equiv. relation is equivalent (isomorphic, if you will) with the equality relation.

Answer 2

Let the relation be called $R$. Then, $(a,b)\in R$ means that the pair $(a,b)$ is in relation $R$, and this fact is also often denoted simply by $aRb$. Now this $R$ relation is just the equality relation on the base set $\{0,1,2,3\}$.

In general, if $R$ is an equivalence relation, an element $x$ belongs to the equivalence class of another element $a$ iff $x$ and $a$ are 'equivalent according to $R$', that is, $aRx$, in other words, $(a,x)\in R$.