Final answer:
The statement ∀x∀yP(x, y) is true when P(x, y) is a universal predicate true for all x and y, and false if there's at least one x, y pair for which P(x, y) is false.
Step-by-step explanation:
The statement ∀x∀yP(x, y) is a universal statement in predicate logic. It is making an assertion about all possible values of x and y in the predicate P.
True scenarios for the statement ∀x∀yP(x, y):
- (a) True, when there is a universal predicate for all x and y
- (c) True, when P(x, y) is true for all x and y
False scenarios for the statement ∀x∀yP(x, y):
- (b) False, when there exists at least one x and y for which P(x, y) is false
- (d) False, when P(x, y) is true for at least one x and y (because the statement claims that P(x, y) must be true for all x and y, not just one)
Thus, statements (a) and (c) describe when it is true, and statements (b) and (d) describe when it is false, albeit with (d) being misleading as it does not capture the universal nature of the initial statement.