Ex 1.4, 2 - Determine whether * is commutative - Class 12 NCERT - Whether binary commutative/associative or not

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  1. Chapter 1 Class 12 Relation and Functions
  2. Serial order wise
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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 Since a * b ≠ b * a * is not 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. (ii) On Q, define a * b = ab + 1 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) * 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) Since (a * b) * c ≠ a * (b * c) * is not an associative binary operation

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.
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