Parallelism is Symmetric Relation
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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 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
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 $\text{A}.2$: $7 \ \text{(a)}$