Last updated at Aug. 11, 2021 by Teachoo

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Ex 1.1, 9 (Introduction) 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} is an equivalence relation. Find the set of all elements related to 1 in each case. Modulus function |1| = 1 |2| = 2 |0| = 0 |−1| = 1 |−3| = 3 Ex 1.1, 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} is an equivalence relation. Find the set of all elements related to 1 in each case. 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 multiple of 4} Note: Multiples of 4 are 0, 4, 8, 12 i.e. |a – b| can be 0, 4, 8, 12 only Check reflexive Since |a – a| = |0| = 0 & 0 is a multiple of 4 So, |a – a| is a multiple of 4 ∴ (a, a) ∈ R, ∴ R is reflexive. Check symmetric We know that |a – b| = |b – a| Hence, if |a – b| is a multiple of 4, then |b – a| is also a multiple of 4 Hence, If (a, b) ∈ R, then (b, a) ∈ R ∴ R is symmetric |a – b| = |–(b – a)| = |b – a| Check transitive If |a – b| is a multiple of 4, then (a – b) is a multiple of 4 Similarly, if |b – c| is a multiple of 4 , then (b – c) is a multiple of 4 Now, Sum of multiple of 4 is also a multiple of 4 a – b + b – c is a multiple of 4 ⇒ a – c is a multiple of 4 Hence, |a – c| is a multiple of 4 ∴ If |a – b| &|b – c| is a multiple of 4 , then |a – c| is a multiple of 4 i.e. If (a, b) ∈ R & (b, c) ∈ R , then (a, c) ∈ R ∴ R is transitive Rough 8 & 12 are multiple of 4 8 + 12 = 20 is also a multiple of 4 Since R is reflexive, symmetric and transitive, it is equivalence relation We need to find set of elements related to 1 R = {(a, b) : |a – b| is a multiple of 4} & A = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} Here a = 1, The set of elements related to 1 are {1, 5, 9}

Ex 1.1

Ex 1.1, 1 (i)

Ex 1.1, 1 (ii)

Ex 1.1, 1 (iii) Important

Ex 1.1, 1 (iv)

Ex 1.1, 1 (v)

Ex 1.1, 2

Ex 1.1, 3

Ex 1.1, 4

Ex 1.1, 5 Important

Ex 1.1, 6

Ex 1.1, 7

Ex 1.1, 8

Ex 1.1, 9 (i) You are here

Ex 1.1, 9 (ii)

Ex 1.1, 10 (i)

Ex 1.1, 10 (ii)

Ex 1.1, 10 (iii) Important

Ex 1.1, 10 (iv)

Ex 1.1, 10 (v)

Ex 1.1, 11

Ex 1.1, 12 Important

Ex 1.1, 13

Ex 1.1, 14

Ex 1.1, 15 (MCQ) Important

Ex 1.1, 16 (MCQ)

Chapter 1 Class 12 Relation and Functions (Term 1)

Serial order wise

About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.