Equivalence of Definitions of Left Quasi-Reflexive Relation
Theorem
Let $\RR \subseteq S \times S$ be a relation in $S$.
The following definitions of the concept of Left Quasi-Reflexive Relation are equivalent:
Definition 1
$\RR$ is left quasi-reflexive if and only if:
- $\forall x, y \in S: \tuple {x, y} \in \RR \implies \tuple {x, x} \in \RR$
Definition 2
$\RR$ is left quasi-reflexive if and only if:
- $\forall x \in \Dom \RR: \tuple {x, x} \in \RR$
where $\Dom \RR$ denotes the domain of $\RR$.
Proof
$(1)$ implies $(2)$
Let $\RR$ be a left quasi-reflexive relation by definition $1$.
Then by definition:
- $\forall x, y \in S: \tuple {x, y} \in \RR \implies \tuple {x, x} \in \RR$
Let $x \in \Dom \RR$ be arbitrary.
Then by definition of domain:
- $\exists y \in S: \tuple {x, y} \in \RR$
Hence by hypothesis:
- $\tuple {x, x} \in \RR$
As $y$ is arbitrary:
- $\forall x \in \Dom \RR: \tuple {x, x} \in \RR$
Thus $\RR$ is a left quasi-reflexive relation by definition $2$.
$\Box$
$(2)$ implies $(1)$
Let $\RR$ be a left quasi-reflexive relation by definition $2$.
Then by definition:
- $\forall x \in \Dom \RR: \tuple {x, x} \in \RR$
Let $x, y \in S$ be arbitrary such that $\tuple {x, y} \in \RR$.
Then by definition of image set:
- $x \in \Dom \RR$
Hence by hypothesis:
- $\tuple {x, x} \in \RR$
As $\tuple {x, y} \in \RR$ is arbitrary:
- $\forall x, y \in S: \tuple {x, y} \in \RR \implies \tuple {x, x} \in \RR$
Thus $\RR$ is a left quasi-reflexive relation by definition $1$.
$\blacksquare$