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 True Statement is Implied by Every Statement:


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