Equivalence of Definitions of Symmetric Relation

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

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

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

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:

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:

Hence $\RR$ is symmetric by definition 1.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 10$: Equivalence Relations: 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 $3.14 \ \text{(a)}$