Answer
$(a)True$
$(b)True$
$(c)True$
$(d)False$
Work Step by Step
a) $\exists_{0} x\left(x^{2}=-1\right)$ means that there exist no real number x such that $x^{2}=-1$
This statement is true because the square of a real number is always non-negative and thus cannot be equal to $-1$.
b) $\exists_{1} x(|x|=0)$means that there exists exactly one real number $x$ such that
This statement is true because the only real number that has an absolute value of $0$ is the value of $x=0$.
c) $\exists_{2} x\left(x^{2}=2\right) $ Means that there exist exactly two real numbers x such that $x^{2}=2$.
This statement is true because the real numbers whose square is equal to $2, -\sqrt{2} \text { and } \sqrt{2}$ (which are exactly two real numbers).
d) $\exists_{3} x(x=|x|)$ means that there are exactly three real number $x$ such that $x=|x|$
this statement is false because the all positive real numbers have the property that their absolute value is equal to the real number itself and there are infinitely many positive real number (which is thus but more than $3$).
