NCERT Class 12 Maths · Chapter 1 Relations And Functions · Exercise 1.1 Question 10

Question

Give an example of a relation that is (i) symmetric but neither reflexive nor transitive; (ii) transitive but neither reflexive nor symmetric; (iii) reflexive and symmetric but not transitive; (iv) reflexive and transitive but not symmetric; (v) symmetric and transitive but not reflexive.

Accepted Answer

Final Answer: Examples on A={1,2,3} are given in the solution. Step 1: Use the common carrier A={1,2,3}. (i) R={(1,2),(2,1)} is symmetric, lacks (1,1), and fails transitivity on 1R2R1. Step 2: (ii) R={(1,2)} has no composable two-pair chain, so is transitive; it lacks diagonals and the reverse (2,1). Step 3: (iii) R={(1,1),(2,2),(3,3),(1,2),(2,1),(2,3),(3,2)} is reflexive and symmetric. It fails transitivity because (1,2),(2,3) belong but (1,3) does not. Step 4: (iv) R={(1,1),(2,2),(3,3),(1,2)} is reflexive and transitive: composing the lone non-diagonal pair with a diagonal adds no new pair. It is not symmetric. Step 5: (v) R={(1,1)} is symmetric and transitive, but (2,2) is missing, so it is not reflexive on A.

NCERT Class 12 Maths · Chapter 1 Relations And Functions · Exercise 1.1 Question 10

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