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

$\mathcal R$ is coreflexive if and only if:

$\forall x, y \in S: \left({x, y}\right) \in \mathcal R \implies x = y$

Definition 2

$\mathcal R$ is coreflexive if and only if:

$\mathcal R \subseteq \Delta_S$

where $\Delta_S$ is the diagonal relation.


Proof

Definition 1 implies Definition 2

Let $\mathcal R$ be a relation which fulfils the condition:

$\forall x, y \in S: \left({x, y}\right) \in \mathcal R \implies x = y$


Then:

\(\displaystyle \left({x, y}\right)\) \(\in\) \(\displaystyle \mathcal R\)
\(\displaystyle \implies \ \ \) \(\displaystyle \left({x, y}\right)\) \(=\) \(\displaystyle \left({x, x}\right)\) by hypothesis
\(\displaystyle \implies \ \ \) \(\displaystyle \left({x, y}\right)\) \(\in\) \(\displaystyle \Delta_S\) Definition of Diagonal Relation
\(\displaystyle \implies \ \ \) \(\displaystyle \mathcal R\) \(\subseteq\) \(\displaystyle \Delta_S\) Definition of Subset


Hence $\mathcal R$ is coreflexive by definition 2.

$\Box$


Definition 2 implies Definition 1

Let $\mathcal R$ be a relation which fulfils the condition:

$\mathcal R \subseteq \Delta_S$


Then:

\(\displaystyle \left({x, y}\right)\) \(\in\) \(\displaystyle \mathcal R\)
\(\displaystyle \implies \ \ \) \(\displaystyle \left({x, y}\right)\) \(\in\) \(\displaystyle \Delta_S\) Definition of Subset
\(\displaystyle \implies \ \ \) \(\displaystyle x\) \(=\) \(\displaystyle y\) Definition of Diagonal Relation


Hence $\mathcal R$ is coreflexive by definition 1.

$\blacksquare$