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Binary Operations — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsBinary Operations3 Marks
Question
Check the commutativity and associativity of the following binary operations:
'*' on Q defined by a * b = ab2 for all a, b ∈ Q.
✓

Answer

Commutativity: Let $\text{a, b}\in\text{Q}.$ Then,

a * b = ab2

b * a = ba2

Therefore,

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

Thus, * is not commutative on Q.

Associativity: Let $\text{a, b, c}\in\text{Q}.$ Then,

a * (b * c) = a * (bc2)

= a(bc2)2

= ab2c4

(a * b) * c = (ab2) * c

= ab2c2

Therefore,

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

Thus, * is not associative on Q.

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