# Equivalence of Definitions of Symmetric Relation

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## Contents

## Theorem

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

### Definition 1

$\RR$ is **symmetric** if and only if:

- $\tuple {x, y} \in \RR \implies \tuple {y, x} \in \RR$

### Definition 2

$\RR$ is **symmetric** if and only if it equals its inverse:

- $\RR^{-1} = \RR$

### 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 $\RR$ be a relation which fulfils the condition:

- $\tuple {x, y} \in \RR \implies \tuple {y, x} \in \RR$

Then:

\(\displaystyle \) | \(\) | \(\displaystyle \tuple {x, y} \in \RR\) | |||||||||||

\(\displaystyle \) | \(\leadsto\) | \(\displaystyle \tuple {y, x} \in \RR\) | by hypothesis | ||||||||||

\(\displaystyle \) | \(\leadsto\) | \(\displaystyle \tuple {x, y} \in \RR^{-1}\) | Definition of Inverse Relation | ||||||||||

\(\displaystyle \) | \(\leadsto\) | \(\displaystyle \RR \subseteq \RR^{-1}\) | Definition of Subset |

Hence $\RR$ is symmetric by definition 3.

$\Box$

### Definition 3 implies Definition 2

Let $\RR$ be a relation which fulfils the condition:

- $\RR \subseteq \RR^{-1}$

Then by Inverse Relation Equal iff Subset:

- $\RR = \RR^{-1}$

Hence $\RR$ is symmetric by definition 2.

$\Box$

### Definition 2 implies Definition 1

Let $\RR$ be a relation which fulfils the condition:

- $\RR^{-1} = \RR$

Then:

\(\displaystyle \) | \(\) | \(\displaystyle \tuple {x, y} \in \RR\) | |||||||||||

\(\displaystyle \) | \(\leadsto\) | \(\displaystyle \tuple {x, y} \in \RR^{-1}\) | as $\RR^{-1} = \RR$ | ||||||||||

\(\displaystyle \) | \(\leadsto\) | \(\displaystyle \tuple {y, x} \in \RR\) | Definition of Inverse Relation |

Hence $\RR$ is symmetric by definition 1.

$\blacksquare$

## Sources

- 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Exercise $10.6 \ \text{(b)}$ - 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): Chapter $\text{I}$: Sets and Functions: Relations: Theorem $3$ - 1971: Robert H. Kasriel:
*Undergraduate Topology*... (previous) ... (next): $\S 1.19$: Some Important Properties of Relations: Exercise $5$ - 2000: James R. Munkres:
*Topology*(2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 3$: Relations: Exercise $3.14 \ \text{(a)}$