Is Parallelism Reflexive?

Question

The book I'm reading to teach myself Discrete Maths(Schaumm's Outlines of Theory and Problems of Discrete Mathematics), says that parallelism '||' is not reflexive. That struck me as odd, because I assumed it was.

(An explanation without vector geometry): Two lines are paralell if they have equal gradients. I.e: $M_1 = M_2$.

Now for any given line $A$ with gradient $M_1$, $M_1 = M_1$.

Contrast this with perpendicularity Where $M_1\cdot M_2 = -1$. For a line to be perpendicular to itself, $M_1^2 = -1$. $M_1 = \sqrt{-1}$

$M_1 \not\in R$
$\therefore$ a line is not perpendicular to itself.

So is paralellism reflexive?

Answer 1

It may depend on your definition of parallel.

If it means "lies in the same plane but with no coincident points" then no line is parallel to itself, given each point is coincident with itself

Answer 2

As @Mauro says in the comments, it depends on the definition of parallel. The question that arises is: Can a line be parallel to itself?

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This appears to be a matter of convention. Since the advantage of a positive answer far outweighs the alternative, we can take the definition of parallel lines to be lines which do not have exactly one point in common.