Let $R=\{(x,y): x=y^2\}$ be a relation defined in $\mathbb{Z}$. Is it reflexive, symmetric, transitive or antisymmetric?

Question

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?

Answer 1

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

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

3. 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?

Answer 2

None. $(4,2) \in R, (2,4) \notin R, (2,2) \notin R$. It is anti-symmetric.