webwork questions

Hi Dr. Tom! I have some questions about the webwork assignment:

Section1.4 Rosen7: Problem 2

The notation ∃!xP(x)  denotes the proposition " There exists a unique x such that P(x) is true. "

If the universe of discourse is the set of integers, what are the truth values of the following?

∃!x(x>1): my answer is true, but the correct answer is false?

<the critical issue here is that there should exist one and only one solution of the inequality.  If there is no solution, or if there are more than one solutions, then the answer is false.  So, over the integers, how many solutions are there?>

∃!x(x^2=1): my answer is true( 1^2=1?), the correct answer is false?

<Again, the issue is how many solutions are there to the equation x^2=1 over the integers?>

Section1.5 Rosen7: Problem 1

Determine the truth value of the following statements if the universe of discourse of each variable is the set of real numbers:

∃x∀y≠0(xy=1): my answer is true, but the correct answer is false?

<The critical issue here is understanding what the order of quantifiers tells us: ∃x∀y≠0 says that there is some x such that for all y the predicate function xy=1 is true--you have to choose the x and then show that the equation is true for all y≠0.  On the other hand ∀y≠0∃x would say that for every y you could find some x--which is a less restrictive statement>