Parallelism is Symmetric Relation

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 symmetric relation on $S$.

Proof

Let $l_1 \parallel l_2$.

By definition of parallel lines, $l_1$ does not meet $l_2$ when produced indefinitely.

Hence $l_2$ similarly does not meet $l_1$ when produced indefinitely.

That is:

$l_2 \parallel l_1$

Thus $\parallel$ is seen to be symmetric.

$\blacksquare$

Also see

• Parallelism is Reflexive Relation
• Parallelism is Transitive Relation

• Parallelism is Equivalence Relation

Sources

• 1964: Steven A. Gaal: Point Set Topology ... (previous) ... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets
• 1977: Gary Chartrand: Introductory Graph Theory ... (previous) ... (next): Appendix $\text{A}.2$: Cartesian Products and Relations: Problem Set $\text{A}.2$: $7 \ \text{(a)}$