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A relation $$R$$ on a set $$A$$ is an equivalence relation if it satisfies:
- Reflexivity: $$\forall a \in A,\ (a,a) \in R$$
- Symmetry: $$\forall a,b \in A,\ (a,b) \in R \implies (b,a) \in R$$
- Transitivity: $$\forall a,b,c \in A,\ (a,b) \in R \,\wedge\, (b,c) \in R \implies (a,c) \in R$$
The equivalence class of an element $$a\in A$$ is
$$[a] = \{\, x \in A \mid (a,x) \in R \}\,. $$
