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Let $L$ be the set of all the straight lines in the plane. Let $G$ and $H$ be the following relations in $L$: $G =\{(l_1 ,l_2) :l_1\ \text{is parallel to}\ l_2\}$,

My attempt:

The symmetric property is true since if $ l_1 $ is parallel to $ l_2 $, then $ l_2 $ is parallel to $ l_1 $. And the same transitive, it is easily followed.

But the reflexive is not fulfilled, since a line is not parallel to itself.

James A.
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  • This depends on the definition of parallel. Some authors allow lines to be parallel to themselves and others don't. – CyclotomicField Mar 22 '21 at 22:13
  • I suppose, from the context, that the goal is to prove that $G$ is an equivalence relation on $L$. Note that nowhere in the title or body of the question you make that explicit. Anyway there're only two hypothesis: either you allow a straight line to be parallel to itself, as CyclootomicField suggests, or you can conclude that $G$ is not an equivalence relation. Except you fix the definition by claiming that $l_1Gl_2$ iff $l_1=l_2$ or $l_1$ is parallel to $l_2$. – amrsa Mar 23 '21 at 12:07

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