For any propositions A and B, I'll write ~A to mean the negation of A and A => B to mean "A implies B." Also, If P is a property (or predicate, such as "even" or "mortal"), I'll write P(x) to mean that x satisfies P. Therefore, ~P(x) means that x does not satisfy P.

If A and B are propositions, then ~A => ~B does not in general imply A => B. However, ~A => ~B is equivalent to B => A. The formula ~A => ~B is called the contrapositive of B => A.

Question A says "For every x, ~A(x) => ~B(x)." Using the information above, this is the same as "For every x, B(x) => A(x)," but not the same as "For every x, A(x) => B(x)."

B is in fact false. If X, Y and Z are propositions, then (X or Y) => Z is equivalent ((X => Z) and (Y => Z)), but x ∈ A ∩ B does not imply that x ∈ A ∩ C.

E is true: consider two cases, when x ∈ A and x ∈ B.

G is false: consider x ∈ B, but x ∉ A and x ∉ C.

I'll let someone else to do the rest, or I'll return to this later.