Which of the following statement are truth statements if universe of…
2023
Which of the following statement are truth statements if universe of disclosure is set of integers :
(A) \(\forall n(n^2 \geq 0)\)
(B) \(\exists n(n^2 = 2)\)
(C) \(\forall n(n^2 \geq n)\)
(D) \(\exists n (n^2 < 0)\)
Choose the correct answer from the options given below :
Answer: D. (A) and (C) Only — Answer: The true statements are: 'for all integers n, n^2 ≥ 0' and 'for all integers n, n^2 ≥ n'. Statement: For all integers n, n^2 ≥ 0. Reason: The square…
- A.
(A) and (B) Only
- B.
(B) and (C) Only
- C.
(C) and (D) Only
- D.
(A) and (C) Only
Attempted by 45 students.
Show answer & explanation
Correct answer: D
Answer: The true statements are: 'for all integers n, n^2 ≥ 0' and 'for all integers n, n^2 ≥ n'.
Statement: For all integers n, n^2 ≥ 0. Reason: The square of any integer is never negative, so this holds for every integer n.
Statement: There exists an integer n such that n^2 = 2. Reason: This is false because √2 is not an integer; no integer squared equals 2.
Statement: For all integers n, n^2 ≥ n. Reason: n^2 − n = n(n−1). If n ≥ 1 then both n and n−1 are nonnegative, so the product is nonnegative; if n ≤ 0 then both n and n−1 are nonpositive, so the product is also nonnegative. Thus n^2 ≥ n for every integer n.
Statement: There exists an integer n with n^2 < 0. Reason: This is false because integer squares are always ≥ 0; no integer square is negative.
Conclusion: The two true statements are the ones asserting that every integer square is nonnegative and that every integer satisfies n^2 ≥ n. Therefore the correct selection contains those two statements.