Asymmetric Relation is Antisymmetric

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $\mathcal R$ be an asymmetric relation.


Then $\mathcal R$ is also antisymmetric.


Proof

Let $\mathcal R$ be asymmetric.

Then from the definition of asymmetric:

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

Thus:

$\neg \exists \left({x, y}\right) \in \mathcal R: \left({y, x}\right) \in \mathcal R$

Thus:

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

Thus:

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

is vacuously true.

$\blacksquare$


Sources