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**NCERT mathematics class 12 chapter Relations and Functions Exercise 1.1 Problem 9 Solution**. Solutions for other problems are available at Exercise 1.1 SolutionsExercise 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}