Asymmetric Relation is Antisymmetric

Theorem

Let $\RR$ be an asymmetric relation.


Then $\RR$ is also antisymmetric.


Proof

Let $\RR$ be asymmetric.

Then from the definition of asymmetric:

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

Thus:

$\neg \exists \tuple {x, y} \in \RR: \tuple {y, x} \in \RR$

Thus:

$\set {\tuple {x, y} \in \RR \land \tuple {y, x} \in \RR} = \O$

Thus:

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

is vacuously true.

$\blacksquare$


Sources

  • 1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): Chapter $4$: The Predicate Calculus $2$: $5$ Properties of Relations: $159$
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