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COMEDK2025MathematicsSets and RelationsActual

If R is a relation defined on N the set of all natural numbers defined by R= (x, y) if and only if x divides y , for all x, y N the R is

Options

  1. Atransitive and symmetric
  2. Bequivalence relation
  3. Creflexive and symmetric
  4. Dreflexive and transitive

Correct answer

D. reflexive and transitive

Step-by-step solution

The relation R is defined on the set of natural numbers N such that x R y if and only if x divides y . Reflexivity: For any x N , x divides x because x = x 1 . Thus, (x, x) R for all x N . The relation is reflexive. Symmetry: For the relation to be symmetric, (x, y) R must imply (y, x) R . If x divides y , then y = kx for some k N . If x y , y does not divide x . For example, 2 divides 4 but 4 does not divide 2 . Thus, the relation is not symmetric. Transitivity: If (x, y) R and (y, z) R , then x divides y and y di

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