Definition:Antisymmetric Relation

Definition

Let $\mathcal R \subseteq S \times S$ be a relation in $S$.

$\mathcal R$ is antisymmetric iff:

$\left({x, y}\right) \in \mathcal R \land \left({y, x}\right) \in \mathcal R \implies x = y$

Note the difference between:

An asymmetric relation, in which the fact that $\left({x, y}\right) \in \mathcal R$ means that $\left({y, x}\right)$ is definitely not in $\mathcal R$

and:

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

Also see

• Symmetry

• Symmetric Relation
• Asymmetric Relation
• Non-symmetric Relation

Sources

• Paul R. Halmos: Naive Set Theory (1960)... (previous)... (next): $\S 14$: Order
• Seth Warner: Modern Algebra (1965): $\S 14$
• A.N. Kolmogorov and S.V. Fomin‎: Introductory Real Analysis (1968): $\S 3.1$
• T.S. Blyth: Set Theory and Abstract Algebra (1975): $\S 1$
• Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (1993): $\S 1.5$