The question in my book says:
Determine whether the relation defined on the set of positive integers is reflexive, symmetric, antisymmetric, transitive, and/or a partial order.
$x = y^2 \rightarrow (x,y) \in R$
I thought it was antisymmetric, but also transitive, and symmetric, and not reflexive. My reasoning was that $R = \{(1,1)\}$ because $1$ is the only positive integer that will equal its square. So it is trivially antisymmetric, transitive, and symmetric. While not being reflexive since $(2,2) \notin R$. Have I been mistaken? The answer in the back of my book says only:
Antisymmetric
Best Answer
Consider:
(Transitivity) If $x=y^2$, and $y=z^2$, does that imply that $x=z^2$?
(Symmetry) If $x=y^2$, does that imply $y=x^2$?