Perpendicularity is Antitransitive 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 antitransitive relation on $S$.


Proof

Let $l_1 \perp l_2$ and $l_2 \perp l_3$.

Then $l_1$ and $l_3$ are parallel, and not perpendicular.

Thus $\perp$ is seen to be antitransitive.

$\blacksquare$


Also see


Sources

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