Equivalence Classes of Diagonal Relation

Theorem

Let $S$ be a set.

Let $\Delta_S$ denote the diagonal relation on $S$.

The set $\EE_S$ of equivalence classes of $S$ can be expressed as:

$\EE_S = \set {\set x: x \in S}$

That is, it is the set of all singletons of $S$.

Proof

Let $x \in S$.

Then by definition of the diagonal relation:

$y \mathrel {\Delta_S} x \iff y = x$

Hence:

$y \in \eqclass x {\Delta_S} \iff y = x$

That is:

$\eqclass x {\Delta_S} = \set x$

Hence the result.

$\blacksquare$