Let P and Q be two non-void relations on a set A. Which of the following statements are correct?

I. P and Q are reflexive ⇒ P ∩ Q is reflexive.

II. P and Q are symmetric ⇒ P ∪ Q is symmetric.

III. P and Q are transitive ⇒ P ∩ Q is transitive.

Select the answer using the code given below.

Explanation:

(I) Given that P and Q are reflexive relations of A.

⇒ For any a∈A, (a, a) ∈ P and (a, a) ∈ Q 

⇒ (a, a )∈ P∩Q

⇒ P∩Q is reflexive relation.

So statement (I)is true

(II) Given that P and Q are symmetric relation

Let (a, b) ∈ (P∪Q)

⇒ (a, b) ∈ P or (a, b) ∈ Q.

⇒ (b, a) ∈ P or (b, a) ∈ Q

⇒  (b, a) ∈ (P∪Q)

⇒P∪Q is symmetric relation.

So statement (II) is true.

(III) Given that P and Q are transitive

Let (a, b)∈ (P∩Q) and (b, c)∈ (P∩Q)

⇒ {(a, b)∈ P, (b, c) ∈ P} and {(a, b)∈Q, (b, c) ∈Q}.....     {∵ P and Q are transitive}

⇒ (a, c) ∈ P∩Q

Thus P∩Q is a transitive relation.

So statement (III) is also true.

∴ Option (d) is correct.