Parallelism is Equivalence 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 an equivalence relation on $S$.


Proof

Checking in turn each of the criteria for equivalence:


Reflexivity

We have that for a straight line $l$:

$l \parallel l$

Thus $\parallel$ is seen to be reflexive.

$\Box$


Symmetry

If $l_1 \parallel l_2$ then $l_2 \parallel l_2$.

Thus $\parallel$ is seen to be symmetric.


$\Box$


Transitivity

From Parallelism is Transitive:

$l_1 \parallel l_2$ and $l_2 \parallel l_3$ implies $l_1 \parallel l_3$.

Thus $\parallel$ is seen to be transitive.


$\Box$


$\parallel$ has been shown to be reflexive, symmetric and transitive.

Hence by definition it is an equivalence relation.

$\blacksquare$


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