Question

Assume that the domain of x is 0, 1 and the domain of y is 2, 4, 7. Which of the following is the correct representation of ∀x∃y q(x, y) as a formula without quantification?

1) q(0, 2) ∧ q(0, 4) ∧ q(0, 7) ∧ q(1, 2) ∧ q(1, 4) ∧ q(1, 7)
2) q(0, 2) ∨ q(0, 4) ∨ q(0, 7) ∨ q(1, 2) ∨ q(1, 4) ∨ q(1, 7)
3) q(0, 2) ∧ q(1, 4) ∨ q(0, 7) ∧ q(1, 2) ∨ q(0, 4) ∧ q(1, 7)
4) q(0, 2) ∨ q(1, 4) ∧ q(0, 7) ∨ q(1, 2) ∧ q(0, 4) ∨ q(1, 7)

Answer 1

The correct representation of the statement ∀x∃y q(x, y) for the domains x={0,1} and y={2,4,7} is achieved by using the logical OR operator between q values for each x, corresponding to Option 2 in the question.

Step-by-step explanation:

The question involves translating a statement with quantifiers into a quantifier-free logical formula. The statement ∀x∃y q(x, y) reads as 'For every x, there exists a y such that q(x, y) is true' given the specified domains of x and y. Given the domains of x={0,1} and y={2,4,7}, we need to express that for each value of x there must be at least one corresponding value of y for which q(x, y) is true. This is different from saying that q(x, y) must be true for all combinations of x and y, which would require using the logical AND operator ∧ between all pairs. Instead, we must use the logical OR operator ∨ for the values of y for each x to represent the existence relationship correctly. Therefore, the correct representation of the given statement without quantification is q(0,2) ∧ (q(0,4) ∨ q(0,7)) ∧ (q(1,2) ∨ q(1,4) ∨ q(1,7)), which simplifies to option 2 listed in the question.