Help on Proving Reflexive, Symmetry and Transitivity for xy>= 1, with relation r E Z , xy E integers,IF AND ONLY IF xy >= 1

Question

My working so far that:

Reflexive: Yes as suppose x E in r, we get x^2 >= 1 which true for all so this is true.

Symmetric: I think it is true since xy >= 1 and xy = yx order not important?

Transitive: suppose xy >= 1 and yz>=1 then xz>=1 must hold true ?

Im not sure if im on the right track or need more depth in m proof help appreciated.

Answer 1

Just try $x=z=\frac12$, $y=4$. Btw, $x=\frac12$ or even $x=0$ will also kill refexivity.