Question

If R is a relation in set X of subsets defined as “A is disjoint to B” then examine the reflexivity, symmetricity and transitivity of R.

Answer 1

Set X = set of subsets A relation R in X is defined as 

ARB ⇔ A is disjoint to B ∀ A, B ∈ X 

⇔ A ∩ B = Φ ∀ A, B ∈ X (where Φ is a null set) 

(i) Reflexivity: 

Let A ∈ X 

A ∈ X ⇒ A ∩ A = A 

⇒ A ∩ A ≠ Φ (until A becomes Φ) 

⇒ A is not disjoint to A 

⇒ (A, A) ∉ R ∀ A ∈ X 

R is not reflexive relation. 

(ii) Symmetricity: 

Let A, B ∈ X in this way 

(A, B) ∈ R 

(A, B) ∈ R 

⇒ A is disjoint to B thus A ∩ B = Φ 

⇒ B ∩ A thus, B is disjoint to A (By commutative)

⇒ (B, A) ∈ R 

So, (A, B) ∈ R 

⇒ (B, A) ∈ R ∀ A, B ∈ X 

R is a symmetric relation.

(iii) Transitivity: 

Let A, B, C ∈ X is in this way 

(A, B) ∈ R and (B, C) ∈ R 

A, B ∈ R ⇒ A ∩ B = Φ 

B, C ∈ R ⇒ B ∩ C = Φ 

Then A ∩ C = Φ is not necessary. 

For example: 

A = {1, 2, 3}, B = {4, 5, 6}, C = {1, 2, 7} 

Here A, B ∈ R because A ∩ B = Φ 

B, C ∈R because B ∩ C ⇒ Φ 

But (A, C) ∉ R because A ∩ C = {1, 2} ≠ Φ 

R is not transitive relation. 

From (i), (ii) and (iii) given relation R is symmetric but relative R is not reflexive and transitive. 

Hence Proved.