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))
Choose the correct answer from the options given below:
Pick an option to check yourself, or open the answer below.
Show answer & explanation
Correct answer: D. (A), (B), (C) and (D)
Explanation
A: x = 0. B: x ≥ 0 and −y > 0 give x − y > 0. C: x = 0, y = −1 gives 1 > 0. D: a product of reals is non-zero iff both factors are non-zero. All four are true.More Discrete Structures and Optimisation questions
- If the universe of discourse is the set of integers, which of the following are TRUE? (A) ∀n ∃m (n² < m) (B) ∃n ∀m (n < m²) (C) ∃n ∀m (nm =…
- If N² = N × N, where N is the set of natural numbers, and R is a relation on N² such that ⟨x, y⟩ R ⟨u, v⟩ ⇔ xv = yu, which of the following…
- Which of the following are tautologies? (A) (P → (P → Q)) → (P → Q) (B) ((P → Q) → Q) → (P ∨ Q) (C) ((P ∨ ¬P) → Q) → ((P ∨ ¬P) → R) (D) (Q…
- Let P(x) be the statement "x = x²". If the universe of discourse is the integers, which of the following have truth value TRUE? (A) P(0) (B…
- Which of the following statements are true if the universe of discourse is the set of integers? (A) ∀n (n² ≥ 0) (B) ∃n (n² = 2) (C) ∀n (n²…
- Which of the following graphs (on vertices a–f) are trees? (A) edges: ac, bc, cd, cf, ef (B) edges: ac, af, df, ef, be (C) edges: ab, ac, a…