This page contains the NCERT mathematics class 12 chapter Relations and Functions Exercise 1.1 Problem 8 Solution. Solutions for other problems are available at Exercise 1.1 Solutions
Exercise 1.1 Problem 8 Solution
8. Show that the relation R in the set A = {1, 2, 3, 4, 5} given by R = {(a, b) : |a – b| is even}, is an equivalence relation. Show that all the elements of {1, 3, 5} are related to each other and all the elements of {2, 4} are related to each other. But no element of {1, 3, 5} is related to any element of {2, 4}.
To Check whether R is Reflexive: The relation R in the set A is reflexive if (a, a) ∈ R, for every a ∈ A.
For any element a, we’ve {|a - a| = 0}, which is even.
∴ 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
As we know, for any two elements x, y, we’ve {|x - y| = |y - x|}. So, when {|x - y|} is even then {|y - x|} is also even.
⇒ If (x, y) ∈ R then (y, x) ∈ R
∴ R is 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
When {|a - b|} is even, then {a - b} is also even.
Similarly, when {|b - c|} is even, then {b - c} is also even.
⇒ {a - c = (a - b) + (b - c)} is also even (∵ The sum of two even numbers is even)
⇒ {|a - c|} is also even.
⇒ when (a, b) ∈ R and (b, c) ∈ R then (a, c) ∈ R
∴ R is reflexive, symmetric and transitive. Hence, it is an equivalence relation.
To Show that all the elements of {1, 3, 5} are related to each other
All the elements in {1, 3, 5} are odd
We know that the difference of any two odd numbers is always even.
⇒ The modulus of their difference is also even.
⇒ All the elements of {1, 3, 5} can be related to each other.
To Show that all the elements of {2, 4} are related to each other
All the elements in {2, 4} are even
We know that the difference of any two even numbers is always even
⇒ The modulus of their difference is also even.
⇒ All the elements of {2, 4} can be related to each other.
To show that no element of {1, 3, 5} is related to any element of {2, 4}.
The elements from {1, 3, 5} can not be related to {2, 4} because their combination forms an odd and an even number.
As we know the difference of an odd and even number is always odd
⇒ They can not be related to each other.
