Trivial Relation is Equivalence

Theorem
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: $$\forall x, z \in S: \left({x, z}\right) \in S \times S$$

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


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