Question

20. Determine the truth value of each statement if the domain consists of all real numbers. (4 pts.)

a) ∃𝑥(𝑥3 = −1) b) ∃𝑥(𝑥4 < 𝑥2) c) ∀𝑥((−𝑥)2 = 𝑥2) d) ∀𝑥(2𝑥 > 𝑥)

Answer 1

a) $\exists x$ ($x^3=-1)$

This is true, $x=-1$

b) $\exists x\ (x^4<x^2)$

This is true, $x=\tfrac{1}{2}$ and we have $x^4=\tfrac{1}{16}<\tfrac{1}{4}=x^2$

c) $\forall x\ ((-x)^2=x^2)$

This is true, $(-x)^2=(-x)\cdot(-x)=(-1)\cdot(-1)\cdot x^2=x^2$

d) $\forall x\ (2x>x)$

This is false, if we have $x=-1$ , then $2x=-2<-1=x$