Exercise 14

For a given predicate $P(x)$, 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?