If universe of disclosure are all real numbers, then which of the following…
2023
If universe of disclosure are all real numbers, then 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 :
- A.
(A) and (B) Only
- B.
(A), (C) and (D) Only
- C.
(A), (B) and (D) Only
- D.
(A), (B) ,(C) and (D) Only
Attempted by 33 students.
Show answer & explanation
Correct answer: D
Answer: All four statements are true.
There exists x such that for all y, x + y = y. Take x = 0; then 0 + y = y for every real y, so the statement is true.
For all x, y, if x ≥ 0 and y < 0 then x - y > 0. Since y < 0 implies -y > 0, we have x - y = x + (-y), a sum of a nonnegative and a positive number, which is strictly positive. Thus the statement is true.
There exist x, y with x ≤ 0 and y ≤ 0 and x - y > 0. Example: x = 0 and y = -1 satisfy x ≤ 0 and y ≤ 0, and 0 - (-1) = 1 > 0. So the statement is true.
For all x, y, (x ≠ 0 and y ≠ 0) if and only if xy ≠ 0. Over the real numbers, a product is zero exactly when at least one factor is zero, so the product is nonzero precisely when neither factor is zero. This equivalence holds for all real x and y.