Parallelism is Equivalence Relation/Transitivity
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Theorem
Let $S$ be the set of straight lines in the plane.
For $l_1, l_2 \in S$, let $l_1 \parallel l_2$ denote that $l_1$ is parallel to $l_2$.
Then $\parallel$ is a transitive relation on $S$.
Proof
From Parallelism is Transitive Relation:
- $l_1 \parallel l_2$ and $l_2 \parallel l_3$ implies $l_1 \parallel l_3$.
Thus $\parallel$ is seen to be transitive.
$\blacksquare$
Sources
- 1977: Gary Chartrand: Introductory Graph Theory ... (previous) ... (next): Appendix $\text{A}.2$: Cartesian Products and Relations: Problem Set $\text{A}.2$: $7 \ \text{(a)}$