Answering questions with existential and/or universal quantifiers

Question

Let P(n): 2n = n²

What is P(2) as a statement?

2(2) = 2²

What is P(3) as a statement?

2(3) = 3²

Based off of this, would it be correct to say ∀n P(n), ∃n P(n), both, or neither? Explain. 

Answer:

Only the existential quantification of the predicate P(n) is correct, since there exists only two values, -2 and 2, for which P(n) is true.

Thus ∃n P(n).

Similarly, since ∀n P(n) states that for all values of n, P(x) is true, and since only two values of n makes P(n) true, the universal quantification of the predicate P(n) is a false statement.

Is my answer above both true and an optimal way to answer?

Answer 1

It's a matter of opinion really.

IMO, this is the cleanest way to answer it:

• $ P(2)\implies \exists{n}:P(n)$
• $\neg P(3)\implies\neg\forall{n}:P(n)$