Answer
a) True
b) False
c) True
d) False
Work Step by Step
a) There are two numbers on the reals that would satisfy the equation, $\exists$ $x (x^{2}=2)$ : $\sqrt 2 $ and $-\sqrt 2 $. Thus, it is true.
b) This can never be true because square of a number can never be negative. It is always positive.
c) Given $\forall$ x ($x^{2} +2\geq1)$ $\equiv$ $\forall$ x ($x^{2} \geq-1)$.
Since square of a number can never be negative. So, $\forall x({x}^2 \geq 0)$ implying that $\forall x({x}^2 \geq -1)$ is always true.
d) This is false. Take x=1.
${1}^2 =1$.
