Question

Determine the truth value of each of these statements if the domain of each variable consists of all real numbers. a) ∃x(x2 = −1) b) ∀x(x2 + 2 ≥ 1)

Answer 1

a) according to statement: for $x\in R: \exist x(x^2 = -1)$. As we know, for $\forall x\isin R: x^2 ≥ 0$. So, it means, that statement (a) is false.

b) according to statement: for $\forall x \isin R:(x^2 + 2 ≥ 1)$. As we know, for $\forall x\isin R: (x^2 ≥ 0)\iff (x^2 + 2 ≥ 2)$

That means statement (b) is true.