Trivial Relation is Largest Equivalence Relation

Theorem

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

$\forall \EE \subseteq S \times S: \EE \subseteq \TT$

where $\EE$ denotes a general equivalence relation.

Proof

The trivial relation $\TT$ on $S$ is defined as:

$\TT = S \times S$

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

Let $\EE$ be an arbitrary equivalence relation on $S$.

By definition of relation, $\EE \subseteq S \times S$ and so (trivially) $\EE \subseteq \TT$.

$\blacksquare$

Sources

• 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 7$: Relations