Question

Which of the following are true statements? The domain of all variables is the integers.

a) ∀x∀y (xy = yx)
b) ∃x∃y (xy = yx)
c) ∃x∀y (xy = yx)
d) ∀x∃y (xy = yx)

Answer 1

All the given statements (a), (b), (c), and (d) are true. They all relate to the commutative property of multiplication within the domain of integers, which asserts that xy always equals yx for any integers x and y.

Step-by-step explanation:

The given statements are in the form of logical quantifiers over the domain of integers. Let's evaluate each statement:

1. ∀x∀y (xy = yx): This statement is saying that for all integers x and for all integers y, xy equals yx. This is a true statement because multiplication of integers is commutative.
2. ∃x∃y (xy = yx): This statement is claiming that there exists an integer x and there exists an integer y such that xy equals yx. This is also true, but it's a weaker statement than the first since commutativity always holds for integers in multiplication.
3. ∃x∀y (xy = yx): This statement indicates that there exists a specific integer x such that for all integers y, xy equals yx. Since multiplication is commutative for all integers, this is true for any x in the domain of integers.
4. ∀x∃y (xy = yx): Here, the statement asserts that for every integer x, there exists an integer y such that xy equals yx. Again, this is true as multiplication is commutative regardless of the choice of x and y within the integers.

Given that the domain is all integers and using the property of commutativity of multiplication, statements a), b), c), and d) are all true statements.