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RELATIONS AND FUNCTIONS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsRELATIONS AND FUNCTIONS1 Mark
MCQ
Choose the correct answer out of the given four options.Let * be binary operation defined on R by a * b = 1 + ab ∀ a, b ∈ R. Then the operation * is:
  • ✓
    Commutative but not associative.
  • B
    Associative but not commutative.
  • C
    Neither commutative nor associative.
  • D
    Both commutative and associative.
✓

Answer

Correct option: A.
Commutative but not associative.
We are given that, a * b = 1 + ab ∀ a, b ∈ R

Consider, a * b = ab + 1

= ba + 1

= b * a

Hence, * is a communicative binary operation.

Also, a * (b * c) = a * (bc + 1) $[\because$ b * c = bc + 1$]$

= a(bc + 1) + 1

= a + abc + 1

Now, (a * b) * c = (ab + 1) * c $[\because$ a * b = ab + 1$]$

= (1 + ab)c + 1

= c + abc +1

Now, $\text{a}+\text{abc}+1\neq\text{c}+\text{abc}+1$

$\Rightarrow\ \text{a}\ ^*\ (\text{b}\ ^* \ \text{c})\neq(\text{a}\ ^* \ \text{b})\ ^* \ \text{c}$

Therefore, * is not associative.

Hence, * is communicative but not associative.

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