Let $R=\{(x,y): x=y^2\}$ be a relation defined in $\mathbb{Z}$. Is it reflexive, symmetric, transitive or antisymmetric? Let $R=\{(x,y): x=y^2\}$ be a relation defined in $\mathbb{Z}$. Is it reflexive, symmetric, transitive or antisymmetric).
I'm having most trouble determining if this relation is symmetric, how can I tell?

*

*The relation is not reflexive: $(2,2)\notin R$

*The relation is not symmetric: $(4,2)\in R$, but $(2,4)\notin R$

*The relation is not transitive: $(16,4)\in R$, $(4,2)\in R$, but $(16,2)\notin R$
Now, let's see if the relation is antisymmetric. Suppose $(x,y)\in R$ and $(y,x)\in R$. Then $x=y^2$ and $y=x^2$, which implies $x=x^4$. Can you now end proving the relation is antisymmetric?
