Equivalence of Definitions of Coreflexive Relation
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Theorem
The following definitions of the concept of Coreflexive Relation are equivalent:
Definition 1
$\RR$ is coreflexive if and only if:
- $\forall x, y \in S: \tuple {x, y} \in \RR \implies x = y$
Definition 2
$\RR$ is coreflexive if and only if:
- $\RR \subseteq \Delta_S$
where $\Delta_S$ is the diagonal relation.
Proof
Definition 1 implies Definition 2
Let $\RR$ be a relation which fulfils the condition:
- $\forall x, y \in S: \tuple {x, y} \in \RR \implies x = y$
Then:
\(\ds \tuple {x, y}\) | \(\in\) | \(\ds \RR\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \tuple {x, y}\) | \(=\) | \(\ds \tuple {x, x}\) | by hypothesis | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \tuple {x, y}\) | \(\in\) | \(\ds \Delta_S\) | Definition of Diagonal Relation | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \RR\) | \(\subseteq\) | \(\ds \Delta_S\) | Definition of Subset |
Hence $\RR$ is coreflexive by definition 2.
$\Box$
Definition 2 implies Definition 1
Let $\RR$ be a relation which fulfils the condition:
- $\RR \subseteq \Delta_S$
Then:
\(\ds \tuple {x, y}\) | \(\in\) | \(\ds \RR\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \tuple {x, y}\) | \(\in\) | \(\ds \Delta_S\) | Definition of Subset | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds x\) | \(=\) | \(\ds y\) | Definition of Diagonal Relation |
Hence $\RR$ is coreflexive by definition 1.
$\blacksquare$