Chapter 1.5, Problem 24E

Translate each of these nested quantifications into an English statement that expresses a mathematical fact. The domain in each case consists of all real numbers.

a) $\exists x\forall y\left(x+y=y\right)$

b) $\forall x\forall y\left(\left(\left(x\ge 0\right)\wedge \left(y<0\right)\right)\to \left(x-y>0\right)\right)$

c) $\exists x\exists y\left(\left(\left(x\le 0\right)\wedge \left(y\le 0\right)\right)\wedge \left(x-y>0\right)\right)$

d) $\forall x\forall y\left(\left(\left(x\ne 0\right)\wedge \left(y\ne 0\right)\right)\leftrightarrow \left(xy\ne 0\right)\right)$