Definition:Equivalence Relation

A relation on a set $$S$$ which is:


 * reflexive,
 * symmetric and
 * transitive

is called an equivalence relation, or an equivalence, on $$S$$.

When discussing equivalence relations, we also use the symbolism $$x \equiv y \left({\mathcal{R}}\right)$$ for $$\left({x, y}\right) \in \mathcal{R}$$.

The Diagonal Relation
The diagonal relation $$\Delta_S$$ on $$S$$is always an equivalence in $$S$$:


 * Reflexive:

$$\forall x \in S: \left({x, x}\right) \in \Delta_S$$ (from its definition)


 * Symmetric:

$$\forall x, y \in S: \left({x, y}\right) \in \Delta_S$$

$$\Longrightarrow x = y$$ (from its definition)

$$\Longrightarrow \left({y, x}\right) \in \Delta_S$$ (from its definition)


 * Transitive:

$$\forall x, y, z \in S: \left({x, y}\right) \in \Delta_S \land \left({y, z}\right) \in \Delta_S$$

$$\Longrightarrow x = y \land y = z$$ (from its definition)

$$\Longrightarrow x = z$$

$$\Longrightarrow \left({x, z}\right) \in \Delta_S$$ (from its definition)

The Trivial Relation
The trivial relation on $$S$$:

$$\mathcal{R} = S \times S$$

is always an equivalence in $$S$$.

Proof:


 * Reflexive: $$\forall x \in S: \left({x, x}\right) \in S \times S$$


 * Symmetric: $$\forall x, y \in S: \left({x, y}\right) \in S \times S \land \left({y, x}\right) \in S \times S$$


 * Transitive: By definition: $$\forall x, y, z \in S: \left({x, z}\right) \in S \times S$$

Thus from "If something is true then anything implies it":

$$\left({x, y}\right) \in S \times S \land \left({x, z}\right) \in S \times S \Longrightarrow \left({x, z}\right) \in S \times S$$