I am a little confused by this relation
R3 is a subset of Z×Z defined by (x,y) in the set R3 if and only if x>2y is it reflexive? Symmetric? antisymmetric? or transitive?
i say its NOT reflexive because (1,2) is not in the set R3
i say it is NOT Symmetric because (1,2) is not in the set(2,1)
i say it is NOT Transitive either because if (4,1) in the set R3, (1,4) is not in the set R3
$R$ is not reflexive, but your reasoning is off. For any non-negative integer, we have that $x \not\gt 2x$. Hence, in those cases, $(x, x) \notin R$.
$R$ is not symmetric, since because $3 \gt 2\cdot 1 = 2$, $(3, 1) \in R$, but since $1 \not\gt 2\cdot 3 = 6, \;(1, 3) \notin R$.
However, $R$ is transitive. If $x \gt 2y$ and $y \gt 2z$, then $x \gt 2y \iff x \gt 2(2z)$, so certainly, $\frac 12 x \gt 2z \implies x \gt 2z$.