Question

If (x, y) ∈ N × N, then xy = x² is a relation that is

(A) Symmetric

(B) Reflexive

(C) Transitive

(D) Equivalence

Answer 1

(D) Equivalence

Let x ∈ R, then xx = x²

∴ x is related to x.

∴ Given relation is reflexive.

Let x = 0 and y = 2,

then xy = 0 × 2 = 0 = x²

∴ x is related to y.

Consider, yx = 2 × 0 = 0 ≠ y²

∴ y is not related to x.

∴ Given relation is not symmetric.

Let x be related to y and y be related to z.

∴ xy = x² and yz = y²

∴ x = \(\frac{x^2}y\) and z = \(\frac{y^2}y\) = y …..[if y ≠ 0]

Consider, xz = \(\frac{x^2}y\) × y = x²

∴ x is related to z.

∴ Given relation is transitive.