Parallelism is Reflexive 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 reflexive relation on $S$.
Proof
By definition of parallel lines, the contemporary definition is for a straight line to be declared parallel to itself.
Hence for a straight line $l$:
- $l \parallel l$
Thus $\parallel$ is seen to be reflexive.
$\blacksquare$
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 $\text{A}.2$: $7 \ \text{(a)}$