Let $R$ be the relation on $\mathbb{Z} \times \mathbb{Z}$ defined by $(w, x)R(y,z)$ if and only if $w + x \leq y + z$. Let $S$ be the relation on $\mathbb{Z} \times \mathbb{Z}$ defined by $(w, x)S(y,z)$ if and only if $w \leq y$ and $x \leq z$.
(a) Is $R$ antisymmetric?
(b) Is $S$ antisymmetric?
I said no for $R$ because it is symmetric therefore cannot be antisymmetric and yes for $S$ because $(w,x)$ must always be less then or equal to $(y,z)$. Just wanted to make sure that I was correct, or am on the right track.
If $(w,x)R(y,z)$ then $w+x \leq y+z$. For example, $(1,2)R(3,4)$. Suppose you want to check the symmetry in this case: you should get that this implies $(3,4)R(1,2)$, that is $3+4\leq 1+2$. This is clearly false. So $R$ is not symmetric.
Your discussion on $S$ looks weak: assume $(w,x)S(y,z)$ and $(y,z)S(w,x)$. Is it true that $(w,x) = (y,z)$? Well, the first means $w \leq y$ and $x \leq z$. The second $y \leq w$ and $z \leq x$. So $w \leq y \leq w$ and $x \leq z \leq x$, hence...