Answer
a)$\exists x \exists y [(x<0 \land y<0)\rightarrow xy>0)]$
b)$\forall x (x-x=0)$
c) $\forall x [x>0 \rightarrow \exists y \exists z(x=y^{2} \land x=z^{2} \land y\ne z)]$
d)$\forall x [x<0 \rightarrow \neg \exists y (x=y^{2})]$
Work Step by Step
a) Rewrite statement:
If a real number is negative and
another real number is also negative
then the product is positive.
$$\exists x \exists y [(x<0 \land y<0)\rightarrow xy>0)]$$
b)Rewrite statement:
The difference of every real number
and itself equals zero.
$$\forall x (x-x=0)$$
c) Rewrite statement:
If a real number is positive then there
exist two (different) square roots.
$$\forall x [x>0 \rightarrow \exists y \exists z(x=y^{2} \land x=z^{2} \land y\ne z)]$$
d)Rewrite statement:
If a real number is negative, then it does not have a (real) square root.
$$\forall x [x<0 \rightarrow \neg \exists y (x=y^{2})]$$
