If the universe of discourse is all real numbers, which of the following are true? (A) ∃x∀y (x + y = y) (B) ∀x∀y (((x ≥ 0) ∧ (y < 0)) → (x − y > 0)) (C) ∃x∃y (((x ≤ 0) ∧ (y ≤ 0)) ∧ (x − y > 0)) (D) ∀x∀y (((x ≠ 0) ∧ (y ≠ 0)) ↔ (xy ≠ 0))
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Correct answer: D. (A), (B), (C) and (D) Only
Explanation
(A) x = 0 works. (B) a non-negative minus a negative is positive. (C) x = 0, y = −1. (D) a product of reals is non-zero exactly when both factors are non-zero. All four are true.
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