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":


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