Q3.Marks: +2.0UGC NET Paper 2: Computer Science 7th Dec 2023 Shift 2
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:
1.(A) and (B) Only
2.(C) and (D) Only
3.(A), (B) and (D) Only
4.(A), (B), (C) and (D) Only✓ Correct
Solution
The correct answer is (A), (B), (C) and (D) Only
EXPLANATION:
(A) ∃x ∀y (x + y = y):
This says there exists some real number x such that for all real numbers y, x + y = y. This statement is true for x = 0. So, (A) is true.
(B) ∀x ∀y (((x ≥ 0) ∧ (y < 0) → (x - y > 0)):
This says for all real x, y that if x is positive (or zero) and y is negative then x - y is positive. This is indeed true because if x is positive and y is negative, subtracting y from x will increase x and thus will be positive. Therefore, (B) is true.
(C) ∃x ∃y (((x ≤ 0) ∧ (y ≤ 0) ∧ (x - y > 0)):
The statement means there are some real numbers x and y such that both x and y are negative (or zero) and the difference x - y > 0. This condition will be satisfied when y is more negative than x. So, (C) is also true.
(D) ∀x ∀y ((x ≠ 0) ∧ (y ≠ 0) ↔ (xy ≠ 0)):
This condition means that for all x and y, x and y can't be zero if and only if their product xy isn't zero. This condition is always true because in the real numbers, the product of two non-zero numbers is never zero. So, (D) is also true.
All options from (A) to (D) are true, so the correct answer is: (A), (B), (C) and (D) Only.