Relation is Reflexive and Coreflexive iff Diagonal

Theorem

Let $S$ be a set.

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

Then $\RR$ is reflexive and coreflexive if and only if:

$\RR = \Delta_S$

where $\Delta_S$ is the diagonal relation.

Proof

Necessary Condition

Let $\RR \subseteq S \times S$ be reflexive and coreflexive.

Then:

\(\ds \RR\)

\(\supseteq\)

\(\ds \Delta_S\)

Definition 2 of Reflexive Relation

\(\ds \RR\)

\(\subseteq\)

\(\ds \Delta_S\)

Definition 2 of Coreflexive Relation

\(\ds \leadsto \ \ \)

\(\ds \RR\)

\(=\)

\(\ds \Delta_S\)

Definition 2 of Set Equality

$\Box$

Necessary Condition

Let $\RR = \Delta_S$

By definition of set equality:

$\Delta_S \subseteq \RR$

and

$\RR \subseteq \Delta_S$

From $\Delta_S \subseteq \RR$ it follows by definition that $\RR$ is reflexive.

From $\RR \subseteq \Delta_S$ it follows by definition that $\RR$ is coreflexive.

$\blacksquare$