Question

Check if the relation R on the set A = {1, 2, 3, 4, 5, 6} defined as R = {(x, y) : y is divisible by x} is (i) symmetric (ii) transitive.

Answer 1

A = {1, 2, 3, 4, 5, 6}

R = {(x, y) : y is divisible by x}

(i) Symmetric

Let (x, y) ∈ R y is divisible by x

∴ x is not necessarily divisible by y

(y, x) ∉ R

e.g., (1, 2) ∈ R

2 is divisible by 1

But 1 is not divisible by 2

(2, 1) ∉ R

Hence, Given Relation is not symmetric.

(ii) Transitive

Let (x, y) ∈ R

y is divisible by x   ...(i)

And (y, z) ∈ R

z is divisible by y   ...(ii)

From equation (i) and equation (ii)

z is divisible by x

∴ (x, z) ∈ R

e.g., (1, 2) ∈ R

2 is divisible by 1   ...(i)

(2, 4) ∈ R

4 is divisible by 2   ...(ii)

From equation (i) and equation (ii)

4 is divisible by 1

(1, 4) ∈ R

Hence, Given Relation is transitive.