Solution

The correct answer is (A), (B) and (C) Only

EXPLANATION:

The properties of the relation R defined as R ↔ xv = yu.

• (A) Reflexive:
  • For a relation to be reflexive, (a, a) must be in the relation for every element a in the set.
  • In this case, we have x = u and y = v. So, (u, u) and (v, v) must be in the relation.
  • Since N is the set of natural numbers, for any natural number u or v, (u, u) and (v, v) are in the relation. Therefore, the relation is reflexive.
• (B) Symmetric:
  • For a relation to be symmetric, if (a, b) is in the relation, then (b, a) must also be in the relation for every pair (a, b). In this case, (x, y) is in the relation if and only if (y, x) is in the relation, as xv = yu implies yu = xv. Therefore, the relation is symmetric.
• (C) Transitive:
  • For a relation to be transitive, if (a, b) and (b, c) are in the relation, then (a, c) must also be in the relation for every triplet (a, b, c). In this case, if xv = yu and yw = zv, then multiplying these equations gives xw = zu, which implies (x, w) is in the relation. Therefore, the relation is transitive.
• (D) Asymmetric:
  • For a relation to be asymmetric, if (a, b) is in the relation, then (b, a) must not be in the relation for any pair (a, b). In this case, since (x, y) is in the relation if and only if (y, x) is also in the relation, the relation cannot be asymmetric.

Based on the analysis: The correct answer is: 4) (A), (B) and (C) Only