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RELATIONS AND FUNCTIONS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsRELATIONS AND FUNCTIONS3 Marks
Question
Let * be the binary operation defined on Q. Find which of the following binary operations are commutative:
  1. a * b = a – b ∀ a, b ∈ Q
  2. a * b = a2 + b2 ∀ a, b ∈ Q
  3. a * b = a + ab ∀ a, b ∈ Q
  4. a * b = (a – b)2 ∀ a, b ∈ Q
✓

Answer

Given that * be the binary operation defined on Q.

  1. a * b = a – b ∀ a, b ∈ Q

= -b + a

= -(b - a)

= -b * a

$\therefore\ \text{a}\ ^*\ \text{b}\neq\text{b}\ ^*\ \text{a}$

Hence, * is  not commutative.

  1. a * b = a2 + b2

= b2 + a2

= b * a

Hence, * is commutative.

  1. We have a * b = a + ab and b * a = b + ab

Clearly, $\text{a}+\text{ab}\neq\text{b}+\text{ab}$

So, * is not communicative.

  1. We have a * b = (a – b)2 ∀ a, b ∈ Q

= (-b + a)2

= {-(b - a)}2

= (b - a)2

=  b * a

Hence, * is communicative.

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