Equivalence of Definitions of Symmetric Relation

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Theorem

The following definitions of the concept of Symmetric Relation are equivalent:

Definition 1

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

$\tuple {x, y} \in \mathcal R \implies \tuple {y, x} \in \mathcal R$

Definition 2

$\mathcal R$ is symmetric if and only if it equals its inverse:

$\mathcal R^{-1} = \mathcal R$

Definition 3

$\mathcal R$ is symmetric if and only if it is a subset of its inverse:

$\mathcal R \subseteq \mathcal R^{-1}$


Proof

Definition 1 implies Definition 3

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

$\left({x, y}\right) \in \mathcal R \implies \left({y, x}\right) \in \mathcal R$


Then:

\(\displaystyle \) \(\) \(\displaystyle \left({x, y}\right) \in \mathcal R\)
\(\displaystyle \) \(\implies\) \(\displaystyle \left({y, x}\right) \in \mathcal R\) by hypothesis
\(\displaystyle \) \(\implies\) \(\displaystyle \left({x, y}\right) \in \mathcal R^{-1}\) Definition of Inverse Relation
\(\displaystyle \) \(\implies\) \(\displaystyle \mathcal R \subseteq \mathcal R^{-1}\) Definition of Subset


Hence $\mathcal R$ is symmetric by definition 3.

$\Box$


Definition 3 implies Definition 2

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

$\mathcal R \subseteq \mathcal R^{-1}$

Then by Inverse Relation Equal iff Subset:

$\mathcal R = \mathcal R^{-1}$

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

$\Box$


Definition 2 implies Definition 1

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

$\mathcal R^{-1} = \mathcal R$


Then:

\(\displaystyle \) \(\) \(\displaystyle \left({x, y}\right) \in \mathcal R\)
\(\displaystyle \) \(\implies\) \(\displaystyle \left({x, y}\right) \in \mathcal R^{-1}\) as $\mathcal R^{-1} = \mathcal R$
\(\displaystyle \) \(\implies\) \(\displaystyle \left({y, x}\right) \in \mathcal R\) Definition of Inverse Relation

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

$\blacksquare$


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