Final answer:
The statement ∀x∃y (xy > 0) is false because it does not hold when x is 0; multiplying 0 by any number y will never result in a product greater than zero.
Step-by-step explanation:
The expression ∀x∃y (xy > 0) translates to "for all x, there exists a y such that the product of x and y is greater than zero." To determine the truth value of this expression, consider that for any real number x, if x is positive, we can choose y to also be positive (e.g., y=1), and if x is negative, we can choose y to be negative (e.g., y=-1) to make the product xy positive. However, if x is 0, no value of y will make the product greater than 0, as 0 multiplied by any number is always 0. Therefore, the statement is false because it does not hold for x=0.