Perpendicularity is Antireflexive Relation

Theorem

Let $S$ be the set of straight lines in the plane.

For $l_1, l_2 \in S$, let $l_1 \perp l_2$ denote that $l_1$ is perpendicular to $l_2$.

Then $\perp$ is an antireflexive relation on $S$.

Proof

By definition of perpendicular lines, for $l_1$ to be perpendicular to itself would mean it would have to meet itself in a right angle.

This it does not do.

So $l_1 \not \perp l_1$.

Thus $\perp$ is seen to be antireflexive.

$\blacksquare$

Also see

• Perpendicularity is Symmetric Relation
• Perpendicularity is Antitransitive Relation

Sources

• 1964: Steven A. Gaal: Point Set Topology ... (previous) ... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets