This page contains the NCERT mathematics class 12 chapter Relations and Functions Exercise 1.1 Problem 15 Solution. Solutions for other problems are available at Exercise 1.1 Solutions
Exercise 1.1 Problem 15 Solution
15. Let R be the relation in the set {1, 2, 3, 4} given by R = {(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)}. Choose the correct answer.
I.
R is reflexive and symmetric but not transitive.
II.
R is reflexive and transitive but not symmetric.
III.
R is symmetric and transitive but not reflexive.
IV.
R is an equivalence relation.
Given that R is the relation in the set {1, 2, 3, 4} given by R = {(1, 2), (2, 2), (1, 1), (4,4), (1, 3), (3, 3), (3, 2)}
To Check whether R is Reflexive: The relation R in the set A is reflexive if (a, a) ∈ R, for every a ∈ A.
To be reflexive, for every element a ∈ A we should have (a, a) ∈ R
We see that for every a ∈ {1, 2, 3, 4}, we have (1, 1) ∈ R, (2, 2) ∈ R, (3, 3) ∈ R and (4, 4) ∈ R
∴ R is reflexive.
To Check whether R is Symmetric: The relation R in the set A is symmetric if (a_1, a_2) ∈ R implies that (a_2, a_1) ∈ R, for all a_1, a_2 ∈ A
For a relation to be symmetric, if (a, b) ∈ R then (b, a) ∈ R
We see that (1, 2) ∈ R but (2, 1) ∉ R
∴ R is not symmetric.
To Check whether R is Transitive: The relation R in the set A is transitive if (a_1, a_2) ∈ R and (a_2, a_3) ∈ R implies that (a_1, a_3) ∈ R, for all a_1, a_2, a_3 ∈ A
In this case, (1, 3) ∈ R, (3, 2) ∈ R and also (1, 2) ∈ R
∴ R is transitive.
∴ R is reflexive and transitive by not symmetric. So, option B is the correct choice.
Note:As we know that the relation R is not symmetric, we can choose that the choice B is the correct one and need not check whether R is transitive or not. This is because, only answer B considers that R is not symmetric
