Hello, captainjapan!

Your quantifiers didn't show up.

I'll have to make some guesses . . .

We have: . . . . Truemeans " ", where and are integers.

Determine the truth value of the statements.

We have: . . . . True

We have:

. . There exists such that: . . . . True

This says: for all and . . . not true.

[Find your own counterexample.]

There is an and such that: . . . True

[See parts (a) and (b).]

This says: for any , there is an integer such that . . . not true.

If , we have: . ... not an integer.

This says: for any , there is an integer such that . . . not true.

If , we have: . ... not an integer.