kenneth h rosen chapter 1 section nested quantifers excercise 1.5 question 40

Question

Find a counterexample, if possible, to these universally
quantified statements, where the domain for all variables
consists of all integers.
a) ∀x∃y(x = 1/y)
b) ∀x∃y(y^2 − x < 100)

Comments

x, y $\in$ Z

a) Any x such that x $\neq$ 1 $\land$ x $\neq$ -1, is a counter example for statement a.

y = 1 / x, y $\notin$ Z for x $\neq$ 1, -1

b) Any x such that x < -100, is a counter example for statement b.

y^2 - x < 100 $\implies$ y < $\sqrt{100+x}$, y $\notin$ Z for x < 100