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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

- \([1] = [3] = \{1, 3\}\)
- \([2] = [4] = \{2, 4\}\)