Trivial Relation is Largest Equivalence Relation

Theorem
The trivial relation $\mathcal T$ on $S$ is the largest equivalence in $S$, in the sense that:


 * $\forall \mathcal E \subseteq S \times S: \mathcal E \subseteq \mathcal T$

where $\mathcal E$ denotes a general equivalence relation.

Proof
The trivial relation $\mathcal T$ on $S$ is defined as:
 * $\mathcal T = S \times S$

It is confirmed from Trivial Relation Equivalence that the trivial relation is in fact an equivalence relation.

Let $\mathcal E$ be an arbitrary equivalence relation on $S$.

By definition of relation, $\mathcal E \subseteq S \times S$ and so (trivially) $\mathcal E \subseteq \mathcal T$.