Question
Download Solution PDFLet 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.
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFExplanation:
(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.
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