Question

On the set N of all natural numbers, define the relation R by a R b, if GCD of a and b is 2. Then, R is

विकल्प

• reflexive, but not symmetric

• symmetric only

• reflexive and transitive

• not reflexive, not symmetric, not transitive

Answer 1

symmetric only

Explanation:

• Let a ∈N. Then,

(a, a) ∉R as the GCD of 'a" and 'a' not 2.

R is not reflexive

• Let a, b ∈N. Then,

(a, b) ∉R ⇒ GCD of 'a' and 'b' is 2

⇒ GCD of 'b' and 'a' is 2

⇒ (b, a) ∈R

∴ R is symmetric

• Let a, b, c ∈N. Then,

(a, b) ∈R and (b, c) ∈ R

⇒ GCD of a and b is 2 and GCD of b and C is 2

⇒ GCD of a and C is 2

R is not transitive

For example, let a = 4, b = 10, c = 12

GCD of (4, 10) = 2

GCD of (10, 12) = 2

But GCD of (4, 12) = 4.