Chapter 1.5, Problem 25E

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) $\exists x\forall y\left(xy=y\right)$

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

c) $\exists x\exists y\left(\left(x^2>y\right)\wedge \left(x<y\right)\right)$

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