Is $x-y \ne 0$ a transitive relation?

Question

I know that for a relation R is transitive if, for all elements aRb and bRc implies aRc.

I came across this question, " Is x-y ≠ 0 transitive?"

I think it is transitive since

x - y ≠ 0

y - z ≠ 0

So we get x - z ≠ 0.

But the answer says that it is not transitive, since it wouldn't be valid for the case where x = z.

Shouldn't a,b and c be distinct?

I am confused as the answer given is essentially building the argument saying that(with the use of only two elements)

x - y ≠ 0

y - x ≠ 0

x - x ≠ 0

Therefore it is not transitive.

Would highly appreciate if you could make it clear if x-y ≠ 0 is transitive or not and why it is so.

Answer 1

I am confused as the answer given is essentially building the argument saying that
(with the use of only two elements).

The arguments is not using just 2 elements, it's using 3 elements.
It's just that the 3rd one ($x$) happens to be the same as the first ($x$).
In the definition for transitivity, no one says the elements taking part should be different.

So the relation is not transitive.

Take any two elements $x,y$ such that $x \ne y$. Now take a third element $z = x$. Test these three elements (against the definition), and you will see that the transitivity does not hold true for these 3 elements.

Answer 2

It's quite simple: let $x \neq y$ in $X$ (the set on which $R$ is defined, which we need to have two points or more, or else $R = \{((x,y) \mid x \neq y\} = \emptyset$ and that is trivially a transitive relation.)

Note that $(x,y) \in R$ and $(y,x) \in R$ but $(x,x) \notin R$, thereby showing $R$ is not transitive. One such example suffices to show that.