Question

Find a counterexample, if possible, to these universally quantified statements, where the domain for all variables consists of all real numbers.

a) ∀x(x2 ( x)

b) ∀x(x2 ( 2)

c) ∀x(|x| > 0

Answer 1

SOLUTION

a) ∀x(x2 ( x)

solution

$x=2,y=1,x\not= \frac{1}1$

b) ∀x(x2 ( 2)

solution

$x=1,y=20, \\20^2-1>100$

c) ∀x(|x| > 0

solution

$x=8,y=4,\\8^2=4^3$