Question 17387 – NTA UGC NET Dec 2023 Paper-2
March 22, 2024NTA UGC NET Dec 2023 Paper-2
March 22, 2024NTA UGC NET Dec 2023 Paper-2
| Question 46 |
If N2 = N x N, N is set of natural numbers and R is relation on N2, s.t. R C N2 x N2 i.e R ⇔ xv=yu, then which of the followings are TRUE
(A) Reflexive
(B) Symmetric
(C) Transitive
(D) Asymmetric
Choose the correct answer from the options given below:
(A) Reflexive
(B) Symmetric
(C) Transitive
(D) Asymmetric
Choose the correct answer from the options given below:
| (A) and (B) Only | |
| (B) and (C) Only | |
| (A), (C) and (D) Only | |
| (A), (B) and (C) Only |
Question 46 Explanation:
(A) Reflexive:
A relation R is reflexive if (a, a) is in R for every element a in the set. In this case, for any natural number N, we have N * N = N^2, so (N, N) is in R. Therefore, the relation is reflexive.
(B) Symmetric:
A relation R is symmetric if whenever (a, b) is in R, then (b, a) is also in R. In this case, if (x, y) is in R, it means that x * y = y * x, so (y, x) is also in R. Therefore, the relation is symmetric.
(C) Transitive:
A relation R is transitive if whenever (a, b) is in R and (b, c) is in R, then (a, c) is also in R. In this case, if (x, y) is in R and (y, u) is in R, it means that x * y = y * x and y * u = u * y. Multiplying these equations, we get x * u = u * x, so (x, u) is in R. Therefore, the relation is transitive.
(D) Asymmetric:
A relation R is asymmetric if whenever (a, b) is in R, then (b, a) is not in R. In this case, since (x, y) is in R implies x * y = y * x, it means that (y, x) is also in R. Therefore, the relation is not asymmetric.
A relation R is reflexive if (a, a) is in R for every element a in the set. In this case, for any natural number N, we have N * N = N^2, so (N, N) is in R. Therefore, the relation is reflexive.
(B) Symmetric:
A relation R is symmetric if whenever (a, b) is in R, then (b, a) is also in R. In this case, if (x, y) is in R, it means that x * y = y * x, so (y, x) is also in R. Therefore, the relation is symmetric.
(C) Transitive:
A relation R is transitive if whenever (a, b) is in R and (b, c) is in R, then (a, c) is also in R. In this case, if (x, y) is in R and (y, u) is in R, it means that x * y = y * x and y * u = u * y. Multiplying these equations, we get x * u = u * x, so (x, u) is in R. Therefore, the relation is transitive.
(D) Asymmetric:
A relation R is asymmetric if whenever (a, b) is in R, then (b, a) is not in R. In this case, since (x, y) is in R implies x * y = y * x, it means that (y, x) is also in R. Therefore, the relation is not asymmetric.
Correct Answer: D
Question 46 Explanation:
(A) Reflexive:
A relation R is reflexive if (a, a) is in R for every element a in the set. In this case, for any natural number N, we have N * N = N^2, so (N, N) is in R. Therefore, the relation is reflexive.
(B) Symmetric:
A relation R is symmetric if whenever (a, b) is in R, then (b, a) is also in R. In this case, if (x, y) is in R, it means that x * y = y * x, so (y, x) is also in R. Therefore, the relation is symmetric.
(C) Transitive:
A relation R is transitive if whenever (a, b) is in R and (b, c) is in R, then (a, c) is also in R. In this case, if (x, y) is in R and (y, u) is in R, it means that x * y = y * x and y * u = u * y. Multiplying these equations, we get x * u = u * x, so (x, u) is in R. Therefore, the relation is transitive.
(D) Asymmetric:
A relation R is asymmetric if whenever (a, b) is in R, then (b, a) is not in R. In this case, since (x, y) is in R implies x * y = y * x, it means that (y, x) is also in R. Therefore, the relation is not asymmetric.
A relation R is reflexive if (a, a) is in R for every element a in the set. In this case, for any natural number N, we have N * N = N^2, so (N, N) is in R. Therefore, the relation is reflexive.
(B) Symmetric:
A relation R is symmetric if whenever (a, b) is in R, then (b, a) is also in R. In this case, if (x, y) is in R, it means that x * y = y * x, so (y, x) is also in R. Therefore, the relation is symmetric.
(C) Transitive:
A relation R is transitive if whenever (a, b) is in R and (b, c) is in R, then (a, c) is also in R. In this case, if (x, y) is in R and (y, u) is in R, it means that x * y = y * x and y * u = u * y. Multiplying these equations, we get x * u = u * x, so (x, u) is in R. Therefore, the relation is transitive.
(D) Asymmetric:
A relation R is asymmetric if whenever (a, b) is in R, then (b, a) is not in R. In this case, since (x, y) is in R implies x * y = y * x, it means that (y, x) is also in R. Therefore, the relation is not asymmetric.
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