Prove or Disprove quantifiers (propositions logic)

Question

What approach can i take to solve these question:
Prove or disprove the following statements. The universe of discourse is N = {1,2,3,4,...}.

(a) ∀x∃y,y = x·x
(b) ∀y∃x,y = x·x
(c) ∃y∀x,y = x·x.

Answer 1

The best way to solve such problems is first to think about them until you're confident that they can be either proven or disproven.

If they can be disproven, then all you have to do to disprove the statement is provide a counterexample. For instance, for b, I can think of the counterexample y=2. There is no number x in N for which n*n = 2. Thus, there is a counterexample, and the statement is false.

If the statement appears to be true, it may be necessary to use some axioms or tautologies to prove the statment. For instance, it is known that two integers that are multiplied together will always produce another integer.

Hopefully this is enough of an approach to get you going.

Answer 2

To prove something exists, find one example for which it is true.
To prove ∀x F(x), take an arbitrary constant a and prove F(a) is true.
Counterexamples can be used to disprove ∀ statements, but not ∃ statements. To disprove ∃x F(x), prove that ∀x !F(x). So, take an arbitrary constant a and show that F(a) is false.