Is the relation $R = \{(x, y) : x = ay\}$ reflexive, symmetric or transitive given a is some rational number

Question

Isn't the following relation supposed to be neither?

I believe for reflexive $x = ax$ holds true for only $a = 1$. But it isn't so for any other rational number. Isn't it that if we can prove any one case where the relation doesn't hold true, it isn't of that type?

I understand it isn't symmetric but how is it supposed to be transitive? Can someone please help me out with an example?

Answer 1

If it is transitive, then we can use $x=ay$,$y=az$ to deduce $x=az$, which implies $a^2=a$, which indicates $a=0$ or $a=1$. But if $a=0$, then the relation is not reflexive.