Equivalence of Definitions of Coreflexive Relation

Theorem

The following definitions of the concept of Coreflexive Relation are equivalent:

Definition 1

$\mathcal R$ is coreflexive if and only if:

$\forall x, y \in S: \left({x, y}\right) \in \mathcal R \implies x = y$

Definition 2

$\mathcal R$ is coreflexive if and only if:

$\mathcal R \subseteq \Delta_S$

where $\Delta_S$ is the diagonal relation.

Proof

Definition 1 implies Definition 2

Let $\mathcal R$ be a relation which fulfils the condition:

$\forall x, y \in S: \left({x, y}\right) \in \mathcal R \implies x = y$

Then:

 $\displaystyle \left({x, y}\right)$ $\in$ $\displaystyle \mathcal R$ $\displaystyle \implies \ \$ $\displaystyle \left({x, y}\right)$ $=$ $\displaystyle \left({x, x}\right)$ by hypothesis $\displaystyle \implies \ \$ $\displaystyle \left({x, y}\right)$ $\in$ $\displaystyle \Delta_S$ Definition of Diagonal Relation $\displaystyle \implies \ \$ $\displaystyle \mathcal R$ $\subseteq$ $\displaystyle \Delta_S$ Definition of Subset

Hence $\mathcal R$ is coreflexive by definition 2.

$\Box$

Definition 2 implies Definition 1

Let $\mathcal R$ be a relation which fulfils the condition:

$\mathcal R \subseteq \Delta_S$

Then:

 $\displaystyle \left({x, y}\right)$ $\in$ $\displaystyle \mathcal R$ $\displaystyle \implies \ \$ $\displaystyle \left({x, y}\right)$ $\in$ $\displaystyle \Delta_S$ Definition of Subset $\displaystyle \implies \ \$ $\displaystyle x$ $=$ $\displaystyle y$ Definition of Diagonal Relation

Hence $\mathcal R$ is coreflexive by definition 1.

$\blacksquare$