For a given predicate \(P(x)\text{,}\) you might believe that the statements \(\forall x P(x)\) or \(\exists x P(x)\) are either true or false. How would you decide if you were correct in each case? You have four choices: you could give an example of an element \(n\) in the domain for which \(P(n)\) is true or for which \(P(n)\) if false, or you could argue that no matter what \(n\) is, \(P(n)\) is true or is false.

1. What would you need to do to prove \(\forall x P(x)\) is true?

2. What would you need to do to prove \(\forall x P(x)\) is false?

3. What would you need to do to prove \(\exists x P(x)\) is true?

4. What would you need to do to prove \(\exists x P(x)\) is false?