Question

On a set N of all natural numbers is defined the relation R by a R b iff the GCD of a and b is 2, then R is

(a) Reflexive and Transitive 

(b) Symmetric and Transitive 

(c) Symmetric only 

(d) Not reflexive, not symmetric, not transitive

Answer 1

(c) Symmetric only

• Let a ∈N. Then 

(a, a) ∉R as the GCD of ‘a’ and ‘a’ is ‘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 

\(\not\Rightarrow\) 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.