Relation is Reflexive Symmetric and Antisymmetric iff Diagonal Relation

Theorem

Let $S$ be a set.

Let $\RR \subseteq S \times S$ be a relation in $S$.

Then:

$\RR$ is reflexive, symmetric and antisymmetric

if and only if:

$\RR$ is the diagonal relation $\Delta_S$.

Proof

Necessary Condition

Let $\RR$ is reflexive, symmetric and antisymmetric.

By definition of reflexive:

$\Delta_S \subseteq \RR$

From Relation is Symmetric and Antisymmetric iff Coreflexive:

$\RR \subseteq \Delta_S$

By definition of set equality:

$\RR = \Delta_S$

$\blacksquare$

Sufficient Condition

Let $\RR = \Delta_S$.

From Relation is Reflexive and Coreflexive iff Diagonal:

$\RR$ is reflexive

and

$\RR$ is coreflexive.

From Relation is Symmetric and Antisymmetric iff Coreflexive it follows that $\RR$ is both symmetric and antisymmetric.

Hence the result.

$\blacksquare$

Sources

• 1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): Chapter $4$: The Predicate Calculus $2$: $5$ Properties of Relations: Exercise $5$