Antisymmetric Relation/Examples

Examples of Antisymmetric Relations

Ordering of Integers

The usual ordering $\le$ on the set of integers $\Z$ is antisymmetric:

$\forall x, y \in \Z: \paren {x \le y} \land \paren {y \le x} \iff x = y$

Set Inclusion

The subset relation is antisymmetric:

$\paren {x \subseteq y} \land \paren {y \subseteq x} \iff x = y$

where $x$ and $y$ are sets.

Partial Ordering

Let $\preccurlyeq$ be a partial ordering on a set $S$.

Then $\preccurlyeq$ is an antisymmetric relation on $S$.