Equivalence of Definitions of Coreflexive Relation

From ProofWiki
Jump to navigation Jump to search

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.

$\blacksquare$