Antireflexive Relation/Examples/Distinctness

Example of Antireflexive Relation

Let $S$ be a set.

Let $\RR$ be the relation on $S$ defined as:

$\forall x, y \in S: x \mathrel \RR y$ if and only if $x$ is distinct from $y$

Then $\RR$ is antireflexive.

Proof

No element of $S$ is distinct from itself.

Hence the result by definition of distinct.

$\blacksquare$

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): irreflexive