Definition:Antisymmetric Relation
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Definition
Let $\RR \subseteq S \times S$ be a relation in $S$.
Definition 1
$\RR$ is antisymmetric if and only if:
- $\tuple {x, y} \in \RR \land \tuple {y, x} \in \RR \implies x = y$
that is:
- $\set {\tuple {x, y}, \tuple {y, x} } \subseteq \RR \implies x = y$
Definition 2
$\RR$ is antisymmetric if and only if:
- $\tuple {x, y} \in \RR \land x \ne y \implies \tuple {y, x} \notin \RR$
Class Theory
In the context of class theory, the definition follows the same lines:
Let $V$ be a basic universe.
Let $\RR \subseteq V \times V$ be a relation in $V$.
$\RR$ is antisymmetric if and only if:
- $\tuple {x, y} \in \RR \land \tuple {y, x} \in \RR \implies x = y$
that is:
- $\set {\tuple {x, y}, \tuple {y, x} } \subseteq \RR \implies x = y$
Also known as
Some sources render antisymmetric relation as anti-symmetric 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
- Results about antisymmetric relations can be found here.
Sources
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): antisymmetric: 1.
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): antisymmetric relation
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): antisymmetric relation
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): antisymmetric relation