Originally Posted by

**dr3amrunn3r** Also. Given an equivalence relation, an equivalence class is formally defined and denoted as the following:

Given an equivalence relation defined on some set A

[a] : {b ϵ A | b R a}

That is, the set of all the elements of A which are related to a.

As an example, consider the equivalence class of nonnegative integers congruent to 1 modulo 2:

[1(mod 2)] = {1, 3, 5, 7, 9, 11, etc} essentially, all the odd integers.

similarly the equivalence class of integers congruent to 0 mod 2:

[0 (mod 2)] = {0, 2, 4, 6, 8, 10, etch} all the even nonnegative integers.

The relation a R b if a is congruent to b modulo 2 is another equivalence relation that might help you with your question. The only equivalence classes of this relation are [0] and [1]. Together, these two equivalence classes partition all of the integers in the set on which this relation is defined.