Definition:Symmetric Relation/Definition 1

Definition

Let $\RR \subseteq S \times S$ be a relation in $S$.

$\RR$ is symmetric if and only if:

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

Also see

• Equivalence of Definitions of Symmetric Relation

• Results about symmetric relations can be found here.

Sources

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• 1955: John L. Kelley: General Topology ... (previous) ... (next): Chapter $0$: Relations
• 1959: E.M. Patterson: Topology (2nd ed.) ... (previous) ... (next): Chapter $\text {II}$: Topological Spaces: $\S 10$. Equivalence Relations
• 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 7$: Relations
• 1964: W.E. Deskins: Abstract Algebra ... (previous) ... (next): $\S 1.2$: Definition $1.5$
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• 1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): Chapter $4$: The Predicate Calculus $2$: $5$ Properties of Relations
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• 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 0.3$
• 1968: A.N. Kolmogorov and S.V. Fomin: Introductory Real Analysis ... (previous) ... (next): $\S 1.4$: Decomposition of a set into classes. Equivalence relations
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• 1971: George E. Andrews: Number Theory: $\S 4.1$: Theorem $4.1$: Remark
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• 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 7$: Relations
• 1977: Gary Chartrand: Introductory Graph Theory ... (previous) ... (next): Appendix $\text{A}.2$: Cartesian Products and Relations
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• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): symmetric relation
• 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 3$: Relations
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• 2008: David Joyner: Adventures in Group Theory (2nd ed.) ... (previous) ... (next): Chapter $2$: 'And you do addition?': $\S 2.3$: Relations: Definition $2.3.2$
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): symmetric relation
• 2012: M. Ben-Ari: Mathematical Logic for Computer Science (3rd ed.) ... (previous) ... (next): Appendix $\text{A}.4$: Definition $\text{A}.21$
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): symmetric relation