Ex 1.4, 11 - Let (a, b) * (c, d) = (a + c, b + d) - Chapter 1

Ex 1.4, 11 - Chapter 1 Class 12 Relation and Functions - Part 2
Ex 1.4, 11 - Chapter 1 Class 12 Relation and Functions - Part 3 Ex 1.4, 11 - Chapter 1 Class 12 Relation and Functions - Part 4

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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

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.