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RELATIONS AND FUNCTIONS — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSRELATIONS AND FUNCTIONS1 Mark
Question
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:
  1. Commutative but not associative.
  2. Associative but not commutative.
  3. Neither commutative nor associative.
  4. Both commutative and associative.
✓

Answer

  1. Commutative but not associative.
Solution:
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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