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?
   Give an example of an element $n$ in the domain for which $P(n)$ is true.
   Give an example of an element $n$ in the domain for which $P(n)$ is false.
   Show for every element $n$ in the domain, that $P(n)$ is true.
   Show for every element $n$ in the domain, that $P(n)$ is false.
2. What would you need to do to prove $\forall x P(x)$ is false?
   Give an example of an element $n$ in the domain for which $P(n)$ is true.
   Give an example of an element $n$ in the domain for which $P(n)$ is false.
   Show for every element $n$ in the domain, that $P(n)$ is true.
   Show for every element $n$ in the domain, that $P(n)$ is false.
3. What would you need to do to prove $\exists x P(x)$ is true?
   Give an example of an element $n$ in the domain for which $P(n)$ is true.
   Give an example of an element $n$ in the domain for which $P(n)$ is false.
   Show for every element $n$ in the domain, that $P(n)$ is true.
   Show for every element $n$ in the domain, that $P(n)$ is false.
4. What would you need to do to prove $\exists x P(x)$ is false?
   Give an example of an element $n$ in the domain for which $P(n)$ is true.
   Give an example of an element $n$ in the domain for which $P(n)$ is false.
   Show for every element $n$ in the domain, that $P(n)$ is true.
   Show for every element $n$ in the domain, that $P(n)$ is false.