Truth Value of Statements with Integers: Proving or Simply Looking?

[Bashyboy]

The question is, "Determine the truth value of each of these statements if the domain consists of all integers?"

The statements are:

$\forall n(n^2 \geq 0)$

$\exists n(n^2=2)$

$\forall n(n^2\geq n)$

$\exists n(n^2 less than 0)$

Does it seem, from reading the question, that I am to determine the truth value of the statement by simply looking at it, or is there some proving process involved?

 

[jbunniii]

If you can determine the truth by simply looking at it, that seems fine to me. If this is homework, you may want to write a brief explanation even if you could see that it was true by inspection. For example, for the first statement you might write "this square of any real number is ______, therefore this statement is _______"

 

[Bashyboy]

What if I was to prove them? How would I go about that? Any hints? Would negating each statement be a good start?

 

[jbunniii]

Negation might be useful for some of these. Others will probably be more straightforward to prove directly.

For example, to prove that $n^2 \geq 0$ is true for all integers $n$, try considering the following three cases separately: $n > 0$, $n = 0$, $n < 0$.