Question
Check the commutativity and associativity of the following binary operations:
'*' on Q defined by a * b = ab2 for all a, b ∈ Q.
'*' on Q defined by a * b = ab2 for all a, b ∈ Q.
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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