determine if the relation $xy \geq 1$ is an equivalence relation for $x,y \in R$

Question

I am having trouble with this one

To me its clear that $xy=yx\geq 1$ symmetric by the associative property.

To me its also clear that $xy = xy$ therefore the relation is reflexive.

we have $xy \geq 1$ and $1\geq\frac{1}{xy}$ so $xy \geq \frac{1}{xy}$ and is thus transitive

Did I correctly show that the relation is both symmetrical and reflexive? Is it not transitive?

Answer 1

For transitivity to hold we need $(xy\ge 1\land yz\ge 1)\implies (xz\ge 1)$ for all $x,y,z.$ But it's false when $x=z=1/2$ and $y=2.$