Final answer:
The statement (∀ y)(∃x) I(x, y) is true and (∃x)(∀ y) I(x, y) is false. The first statement can be satisfied by choosing x equal to y, but there's no single x that can divide all possible y to produce an integer for the second statement.
Step-by-step explanation:
In the domain of nonzero integers, I(x, y) is the predicate which states "x/y is an integer." We are asked to determine the truth values of the following statements given this predicate:
(∀ y)(∃x) I(x, y): This statement is translated as "For every nonzero integer y, there exists a nonzero integer x such that x/y is an integer." This statement is true, as for any nonzero integer y, we can choose x = y (since y is nonzero, x will also be nonzero) which makes x/y = 1, which is an integer.
(∃x)(∀ y) I(x, y): This statement is translated as "There exists a nonzero integer x such that for every nonzero integer y, x/y is an integer." This statement is false, as there is no single nonzero integer x that divided by every nonzero integer y will always result in an integer, due to the fact that not all y will be divisors of a given x.
Examining these statements involves understanding basic number theory and logic pertaining to quantifiers and divisibility within the integers.