# Ex 1.4, 11 - Chapter 1 Class 12 Relation and Functions (Term 1)

Last updated at Jan. 28, 2020 by

Last updated at Jan. 28, 2020 by

Transcript

Ex 1.4, 11 Let A = N × N and * be the binary operation on A defined by (a, b) * (c, d) = (a + c, b + d). Show that * is commutative and associative. Find the identity element for * on A, if any. Check commutative * is commutative if (a, b) * (c, d) = (c, d) * (a, b) ∀ a, b, c, d ∈ R Since (a, b) * (c, d) = (c, d) * (a, b) ∀ a, b, c, d ∈ R * is commutative (a, b) * (c, d) = (a + c, b + d) (c, d) * (a, b) = (c + a, d + b) = (a + c, b + d) (a, b) * (c, d) = (a + c, b + d) Check associative * is associative if (a, b) * ( (c, d) * (x, y) ) = ((a, b) * (c, d)) * (x, y) ∀ a, b, c, d, x, y ∈ R Since (a, b) * ( (c, d) * (x, y) ) = ((a, b) * (c, d)) * (x, y) * is associative (a, b) * ( (c, d) * (x, y) ) = (a, b) * (c + x, d + y) = (a + c + x , b + d + y) ((a, b) * (c, d)) * (x, y) = (a + c, b + d) * (x, y) = (a + c + x , b + d + y) (a, b) * (c, d) = (a + c, b + d) Identity element e is identity of * if (a, b) * e = e * (a, b) = (a, b) where e = (x, y) So, (a, b) * (x, y) = (x, y) * (a, b) = (a, b) (a + x, b + y) = (x + a , b + y) = (a, b) e is the identity of * if a * e = e * a = a Now, (a + x, b + y) = (a, b) Comparing Therefore, the operation * does not have any identity element. a + x = a x = a – a = 0 x = 0 b + y = b y = b – b y = 0 Since A = N × N x & y are natural numbers Since 0 is not natural Identity element does not exist

Ex 1.4

Ex 1.4 ,1 (i)
Deleted for CBSE Board 2022 Exams

Ex 1.4 ,1 (ii) Important Deleted for CBSE Board 2022 Exams

Ex 1.4 ,1 (iii) Deleted for CBSE Board 2022 Exams

Ex 1.4 ,1 (iv) Important Deleted for CBSE Board 2022 Exams

Ex 1.4 ,1 (v) Deleted for CBSE Board 2022 Exams

Ex 1.4, 2 (i) Important Deleted for CBSE Board 2022 Exams

Ex 1.4, 2 (ii) Deleted for CBSE Board 2022 Exams

Ex 1.4, 2 (iii) Deleted for CBSE Board 2022 Exams

Ex 1.4, 2 (iv) Important Deleted for CBSE Board 2022 Exams

Ex 1.4, 2 (v) Deleted for CBSE Board 2022 Exams

Ex 1.4, 2 (vi) Important Deleted for CBSE Board 2022 Exams

Ex 1.4, 3 Deleted for CBSE Board 2022 Exams

Ex 1.4, 4 Deleted for CBSE Board 2022 Exams

Ex 1.4, 5 Deleted for CBSE Board 2022 Exams

Ex 1.4, 6 Important Deleted for CBSE Board 2022 Exams

Ex 1.4, 7 Deleted for CBSE Board 2022 Exams

Ex 1.4, 8 Deleted for CBSE Board 2022 Exams

Ex 1.4, 9 (i) Deleted for CBSE Board 2022 Exams

Ex 1.4, 9 (ii) Deleted for CBSE Board 2022 Exams

Ex 1.4, 9 (iii) Deleted for CBSE Board 2022 Exams

Ex 1.4, 9 (iv) Important Deleted for CBSE Board 2022 Exams

Ex 1.4, 9 (v) Important Deleted for CBSE Board 2022 Exams

Ex 1.4, 9 (vi) Deleted for CBSE Board 2022 Exams

Ex 1.4, 10 Deleted for CBSE Board 2022 Exams

Ex 1.4, 11 Important Deleted for CBSE Board 2022 Exams You are here

Ex 1.4, 12 Deleted for CBSE Board 2022 Exams

Ex 1.4, 13 (MCQ) Important Deleted for CBSE Board 2022 Exams

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.