Question

Let a set A = A 1 ⋃ A 2 ⋃ …⋃ A k, where _A i _⋂ A j = Φ for i ≠ j , 1 ≤ i , j ≤ k. Define the relation R from A to A by R = {(x , y) : y ∈ A i if and only if x ∈ A i, 1 ≤ i ≤ k}. Then, R is

• A. reflexive, symmetric but not transitive
• B. reflexive, transitive but not symmetric
• C. reflexive but not symmetric and transitive
• D. an equivalence relation

Answer 1

Answer: (D)

an equivalence relation

Answer 2

The correct option is(D): an equivalence relation.

R ={(x , y) :y ∈ A i, iff x ∈ A i, 1 ≤ i ≤ k}

(1) Reflexive

(a, a) ⇒ a ∈ Ai iff a ∈ Ai

(2) Symmetric

(a, b) ⇒ a ∈Ai iff b ∈ Ai

(b, a) ∈ R as b ∈ Ai iff a ∈ Ai

(3) Transitive

(a , b) ∈ R & (b, c) ∈ R.

⇒ a ∈ A i iff b ∈ _A i & b _∈ _A i _iff c ∈ A i

⇒ a ∈ A i iff c ∈ A i

⇒ (a , c) ∈ R.

⇒ Relation is equivalence.