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 mutiple of 4}
ii. R = {(a, b) : a = b}
Is an equivalence relation. Find the set of all elements related to 1 in each case

Solution:

i. A = {x ∈ Z : 0 ≤ x ≤ 12} = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}

R = {(a, b): |a - b| is a mutiple of 4}

a ∈ A, (a, a) ∈ R [|a - a| is a mutiple of 4]

Therefore,

R is reflexive.

(a, b) ∈ R ⇒ |a - b| [is a multiple of 4]

⇒ |-(a - b)| = |b - a| [is a multiple of 4]

(b, a) ∈ R

Therefore,

R is symmetric.

(a, b) ∈ R and (b, c) ∈ R

⇒ |a - b| is a multiple of 4 and |b - c| is a multiple of 4

⇒ (a - b) is a multiple of 4 and (b - c) is a multiple of 4

⇒ (a - c) = (a - b) + (b - c) is a multiple of 4

⇒ |a - c| is a multiple of 4

⇒ (a, c) ∈ R

Therefore,

R is transitive.

R is an equivalence relation.

The set of elements related to 1 is {1, 5, 9} as

|1 - 1| = 0 is a multiple of 4.

|5 - 1| = 4 is a multiple of 4.

|9 - 1| = 8 is a multiple of 4.

ii. R = {(a, b): a = b}

a ∈ A, (a, a) ∈ R [since a = a]

Therefore,

R is reflexive.

(a, b) ∈ R

⇒ a = b

⇒ b = a

⇒ (b, a) ∈ R

Therefore,

R is symmetric.

(a, b) ∈ R and (b, c) ∈ R

⇒ a = b and b = c

⇒ a = c

⇒ (a, c) ∈ R

Therefore,

R is transitive.

R is an equivalence relation

The set of elements related to 1 is {1}

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NCERT Solutions for Class 12 Maths - Chapter 1 Exercise 1.1 Question 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 mutiple of 4} ii. R = {(a, b) : a = b}
Is an equivalence relation. Find the set of all elements related to 1 in each case

Summary:

(i). R = {(a, b) : |a - b| is a mutiple of 4} is an equivalence relation. (ii) R = {(a, b) : a = b} Is an equivalence relation.The set of elements related to 1 is {1}