Equivalence of Definitions of Right Quasi-Reflexive Relation

Theorem

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

The following definitions of the concept of Right Quasi-Reflexive Relation are equivalent:

Definition 1

$\RR$ is right quasi-reflexive if and only if:

$\forall x, y \in S: \tuple {x, y} \in \RR \implies \tuple {y, y} \in \RR$

Definition 2

$\RR$ is right quasi-reflexive if and only if:

$\forall y \in \Img \RR: \tuple {y, y} \in \RR$

where $\Img \RR$ denotes the image set of $\RR$.

Proof

$(1)$ implies $(2)$

Let $\RR$ be a right quasi-reflexive relation by definition $1$.

Then by definition:

$\forall x, y \in S: \tuple {x, y} \in \RR \implies \tuple {y, y} \in \RR$

Let $y \in \Img \RR$ be arbitrary.

Then by definition of image set:

$\exists x \in S: \tuple {x, y} \in \RR$

Hence by hypothesis:

$\tuple {y, y} \in \RR$

As $y$ is arbitrary:

$\forall y \in \Img \RR: \tuple {y, y} \in \RR$

Thus $\RR$ is a right quasi-reflexive relation by definition $2$.

$\Box$

$(2)$ implies $(1)$

Let $\RR$ be a right quasi-reflexive relation by definition $2$.

Then by definition:

$\forall y \in \Img \RR: \tuple {y, y} \in \RR$

Let $x, y \in S$ be arbitrary such that $\tuple {x, y} \in \RR$.

Then by definition of image set:

$y \in \Img \RR$

Hence by hypothesis:

$\tuple {y, y} \in \RR$

As $\tuple {x, y} \in \RR$ is arbitrary:

$\forall x, y \in S: \tuple {x, y} \in \RR \implies \tuple {y, y} \in \RR$

Thus $\RR$ is a right quasi-reflexive relation by definition $1$.

$\blacksquare$