Equivalence of Definitions of Quasi-Reflexive Relation

Theorem

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

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

Definition 1

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

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

Definition 2

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

$\forall x \in \Field \RR: \tuple {x, x} \in \RR$

where $\Field \RR$ denotes the field of $\RR$.

Definition 3

$\RR$ is quasi-reflexive if and only if $\RR$ is both left quasi-reflexive and right quasi-reflexive.

Proof

$(1)$ implies $(3)$

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

Then by definition:

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

That is:

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

and:

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

Hence:

$\RR$ is a left quasi-reflexive relation by definition

and:

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

Thus $\RR$ is a quasi-reflexive relation by definition $3$.

$\Box$

$(3)$ implies $(2)$

Let $\RR$ be a quasi-reflexive relation by definition $3$.

Then by definition:

$\RR$ is a left quasi-reflexive relation

and:

$\RR$ is a right quasi-reflexive relation.

Hence by definition of left quasi-reflexive relation:

$\forall x \in \Dom \RR: \tuple {x, x} \in \RR$

and by definition of right quasi-reflexive relation:

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

Let $x \in \Field \RR$ be arbitrary.

Then by definition of field of relation:

$x \in \Dom \RR \cup \Img \RR$

That is, either:

$x \in \Dom \RR$, in which case $\tuple {x, x} \in \RR$

or:

$x \in \Img \RR$, in which case $\tuple {x, x} \in \RR$.

In both cases $\tuple {x, x} \in \RR$.

As $x \in \Field \RR$ is arbitrary:

$\forall x \in \Field \RR: \tuple {x, x} \in \RR$

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

$\Box$

$(2)$ implies $(1)$

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

$\forall x \in \Field \RR: \tuple {x, x} \in \RR$

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

By definition of domain of mapping:

$x \in \Dom \RR$

and so by definition of field of relation:

$x \in \Field \RR$

Thus by hypothesis:

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

Simlarly, by definition of image set of mapping:

$y \in \Img \RR$

and so by definition of field of relation:

$y \in \Field \RR$

Thus by hypothesis:

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

That is:

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

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

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

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

$\blacksquare$