Definition:Antisymmetric Relation/Definition 2

Definition

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

$\RR$ is antisymmetric if and only if:

$\tuple {x, y} \in \RR \land x \ne y \implies \tuple {y, x} \notin \RR$

Also known as

Some sources render antisymmetric relation as anti-symmetric relation.

Some sources (perhaps erroneously) use this definition for an asymmetric relation.

Antisymmetric and Asymmetric Relations

Note the difference between:

An asymmetric relation, in which the fact that $\tuple {x, y} \in \RR$ means that $\tuple {y, x}$ is definitely not in $\RR$

and:

An antisymmetric relation, in which there may be instances of both $\tuple {x, y} \in \RR$ and $\tuple {y, x} \in \RR$ but if there are, then it means that $x$ and $y$ have to be the same object.

Also see

• Equivalence of Definitions of Antisymmetric Relation

• Definition:Symmetry (Relation)

• Definition:Symmetric Relation
• Definition:Asymmetric Relation
• Definition:Non-Symmetric Relation

• Results about antisymmetric relations can be found here.

Sources

• 1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): Chapter $4$: The Predicate Calculus $2$: $5$ Properties of Relations
• 1977: Gary Chartrand: Introductory Graph Theory ... (previous) ... (next): Appendix $\text{A}.2$: Cartesian Products and Relations
• 1993: Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.5$: Relations