# Ex 1.4, 2 - Chapter 1 Class 12 Relation and Functions

Last updated at Dec. 11, 2018 by Teachoo

Last updated at Dec. 11, 2018 by Teachoo

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Ex 1.4, 2 For each binary operation * defined below, determine whether * is commutative or associative. (i) On Z, define a * b = a − b Check commutative * is commutative if a * b = b * a a * b = a – b b * a = b – a Since a * b ≠ b * a * is not commutative Check associative * is associative if (a * b) * c = a * (b * c) (a * b)* c = (a – b) * c = (a – b) – c = a – b – c a * (b * c) = a * (b – c) = a – (b – c) = a – b + c Since (a * b) * c ≠ a * (b * c) * is not an associative binary operation Ex 1.4, 2 For each binary operation * defined below, determine whether * is commutative or associative. (ii) On Q, define a * b = ab + 1 Check commutative * is commutative if a * b = b * a a * b = ab + 1 b * a = ba + 1 = ab + 1 Since a * b = b * a ∀ a, b ∈ Q * is commutative Check associative * is associative if (a * b) * c = a * (b * c) (a * b)* c = (ab + 1) * c = (ab + 1)c + 1 = abc + c + 1 a * (b * c) = a * (bc + 1) = a(bc + 1) + 1 = abc + a + 1 Since (a * b) * c ≠ a * (b * c) * is not an associative binary operation Ex 1.4, 2 For each binary operation * defined below, determine whether * is commutative or associative. (iii) On Q, define a * b = ab/2 Check commutative * is commutative if a * b = b * a Since a * b = b * a ∀ a, b ∈ Q * is commutative Check associative * is associative if (a * b) * c = a * (b * c) Since (a * b) * c = a * (b * c) ∀ a, b, c ∈ Q * is an associative binary operation Ex 1.4, 2 For each binary operation * defined below, determine whether * is commutative or associative. (iv) On Z+, define a * b = 2^𝑎𝑏 Check commutative * is commutative if a * b = b * a Since a * b = b * a ∀ a, b, c ∈ Z+ * is commutative Check associative * is associative if (a * b) * c = a * (b * c) Since (a * b) * c ≠ a * (b * c) * is not an associative binary operation Ex 1.4, 2 For each binary operation * defined below, determine whether * is commutative or associative. (v) On Z+, define a * b = 𝑎^𝑏 Check commutative * is commutative if a * b = b * a Since a * b ≠ b * a * is not commutative Check associative * is associative if (a * b) * c = a * (b * c) Example Let a = 2, b = 3, c = 4 Since (a * b) * c ≠ a * (b * c) * is not an associative binary operation Ex 1.4, 2 For each binary operation * defined below, determine whether * is commutative or associative. (vi) On R − {−1}, define a * b = a/(b + 1) Check commutative * is commutative if a * b = b * a Since a * b ≠ b * a * is not commutative Check associative * is associative if (a * b) * c = a * (b * c) (a * b)* c = (𝑎/(𝑏 + 1)) * c = (𝑎/(𝑏 + 1))/𝑐 = 𝑎/(𝑐(𝑏 + 1)) a * (b * c) = a * (𝑏/(𝑐 + 1)) = 𝑎/(𝑏/(𝑐 + 1)) = (𝑎(𝑐 + 1))/𝑏 Since (a * b) * c ≠ a * (b * c) * is not an associative binary operation

Chapter 1 Class 12 Relation and Functions

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Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.