Question

On the set of natural number N, the relation R is defined by aRb if LCM of a and b is 4. then the relation R is
a) reflexive but not symmetric
b) reflexive and transitive
c) symmetric only
d) neither reflexive not symmetric​

Answer 1

Answer:

LCM of a and a is not equal to 4

For example consider LCM(2,2) =2 it's not equal to 4.

So R is not reflexive

If LCM of and b is 4 then obviously LCM of b and a is also 4. So R is symmetric

If LCM of a and b is 4 and LCM of b and c is also 4 then LCM of a and c cannot be 4

For example consider LCM(1,4)=4 and LCM(4,2)=4 but LCM (1,2) is not equal to 4

So R cannot be transitive

Hence, R is symmetric only

Option c

Answer 2

The correct option is "c"

aRb ⇒ LCM of a and b is 4

It is not Reflexive

If LCM of  a and b is 4 then LCM of b and a is also 4.

(a, a) ∈ R

aRa ⇒ GCD (4, 4) = 4

So R is symmetric

(a, b) ∈ R ⇒ GCD (a, b) = 4

(b, a) ∈ R ⇒ GCD (b, a) = 4

So R is symmetric

Hence the R is symmetric only