Chapter 1.5, Problem 28E

Determine the truth value of each of these statements if the domain of each variable consists of all real numbers.

a) $\forall x\exists y\left(x^2=y\right)$

b) $\forall x\exists y\left(x=y^2\right)$

c) $\exists x\forall y\left(xy=0\right)$

d) $\exists x\exists y\left(x+y\ne y+x\right)$

e) $\forall x\left(x\ne 0\to \exists y\left(xy=1\right)\right)$

f) $\exists x\forall y\left(y\ne 0\to xy=1\right)$

g) $\forall x\exists y\left(x+y=1\right)$

h) $\exists x\exists y\left(x+2y=2\wedge 2x+4y=5\right)$

i) $\forall x\exists y\left(x+y=2\wedge 2x-y=1\right)$

j) $\forall x\forall y\exists z\left(z=\left(x+y\right)/2\right)$