If I define a relation $\rho$ on $\Bbb Z$ by: $x\rho y$ if and only if $x$ is a multiple of $y$. The solution key tells me that this is not an antisymmetric relation.
But say $x$ is a multiple of $y$, then $x=yn$ for some $n\in \Bbb Z$. Then if $y$ is a multiple of $x$, we have $y=xk$ for some $k\in \Bbb Z$.
Hence $x=yn=xnk\implies nk=1$ and hence $x=y$. Which makes this antisymmetric? Is the answer key wrong?
You can't conclude $x=y$ from $nk=1$.
And here is a simple counterexample to it being antisymmetric:
Take $x=1$ and $y=-1$