Equivalence of Definitions of Symmetric Relation
(Redirected from Relation equals Inverse iff Symmetric)
Jump to navigation
Jump to search
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 $3.14 \ \text{(a)}$