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# Equivalence relation

Equivalence relations are a binary relation that is reflexive, symmetric and transitive.

For example $\mathcal R = \{(a,b) \in \mathbb Z^2 \vert a \bmod 2 = b \bmod 2 \}$ represents an equivalence relation on $$\mathbb Z$$.

## Equivalence class

Given a equivalence relation $$\mathcal R$$ on some set $$S$$, the equivalence class of $$a \in S$$ is the subset of $$S$$ containing all elements related to $$a$$ through $$\mathcal R$$. This is denoted with square brackets as below: $[a] = \{x \vert x \in S \text{ and } x \mathcal R a \}$

For the set $$S = \{ 1,2,3,4 \}$$ and example modulo relation above, there are two equivalence classes

1. $$[1] = [3] = \{1, 3\}$$
2. $$[2] = [4] = \{2, 4\}$$