This page contains the NCERT mathematics class 12 chapter Relations and Functions Exercise 1.1 Problem 9 Solution. Solutions for other problems are available at Exercise 1.1 Solutions
Exercise 1.1 Problem 9 Solution
9. Show that each of the relation R in the set A = x ∈ Z : 0 ≤ x ≤ 12, given by
i.
R = {(a, b) : |a – b| is a multiple of 4}
ii.
R = (a, b) : a = b
is an equivalence relation. Find the set of all elements related to R in each case.
To Show that R = {(a, b) : |a – b| is a multiple of 4} is an equivalence relation.
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 a multiple of 4.
∴ 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|}.
⇒ When {|x - y|} is multiple of 4 then |y - x| is also multiple 4 (as both of them are equal).
⇒ 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 multiple of 4, then {a - b} is also a multiple of 4.
Similarly, when {|b - c|} is multiple of 4, then {b - c} is also multiple of 4.
So, we have {a - c = (a - b) + (b - c)} is also multiple of 4 as it is sum of two multiple of 4.
⇒ {|a - c|} is also multiple of 4.
⇒ 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 find the set of elements related to 1 for case (i)
Set A can be written in the Roster form as A = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
We have to find the elements whose difference with 1 is a multiple of 4. In otherwords, {|1 - x|} should be multiple of 4 i.e. 0 or 4 or 8
We have |1 – 1| = 0 which is divisible by 4
We have |1 – 5| = 4 which is divisible by 4
We have |1 – 9| = 8 which is divisible by 4
∴ The set of elements related to 1 are : {1, 5, 9}
To Show that R = {(a, b) : a = b} is an equivalence relation.
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}
⇒ (a, a) ∈ 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
As we know, for any two elements a, b when {a = b} then we’ve {b = a}
⇒ If (a, b) ∈ R then {(b, a)} ∈ 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} and {b = c}, then we’ve {a = c}
⇒ 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 find the set of elements related to 1 for case (ii)
We’ve to find an element such that {1 = x} or {x = 1}
Clearly, 1 is the only element that satisfies this condition.
∴ the set of elements related to 1 is : {1}
