Equivalence of Definitions of Symmetric Relation
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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:
\(\ds \) | \(\) | \(\ds \tuple {x, y} \in \RR\) | ||||||||||||
\(\ds \) | \(\leadsto\) | \(\ds \tuple {y, x} \in \RR\) | by hypothesis | |||||||||||
\(\ds \) | \(\leadsto\) | \(\ds \tuple {x, y} \in \RR^{-1}\) | Definition of Inverse Relation | |||||||||||
\(\ds \) | \(\leadsto\) | \(\ds \RR \subseteq \RR^{-1}\) | 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:
\(\ds \) | \(\) | \(\ds \tuple {x, y} \in \RR\) | ||||||||||||
\(\ds \) | \(\leadsto\) | \(\ds \tuple {x, y} \in \RR^{-1}\) | as $\RR^{-1} = \RR$ | |||||||||||
\(\ds \) | \(\leadsto\) | \(\ds \tuple {y, x} \in \RR\) | Definition of Inverse Relation |
Hence $\RR$ is symmetric by definition 1.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): 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 $3.14 \ \text{(a)}$