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Question
Check the commutativity and associativity of the following binary operation '*' on Q defined by a * b = ab2 for all a, b ∈ Q ?
Solution
Commutativity :
\[\text{Let }a, b \in Q . \text{Then}, \]
\[a * b = a b^2 \]
\[b * a = b a^2 \]
\[\text{Therefore},\]
\[a * b \neq b * a\]
Thus, * is not commutative on Q.
Associativity :
\[\text{Let }a, b, c \in Q . \text{Then}, \]
\[a * \left( b * c \right) = a * \left( b c^2 \right)\]
\[ = a \left( b c^2 \right)^2 \]
\[ = a b^2 c^4 \]
\[\left( a * b \right) * c = \left( a b^2 \right) * c\]
\[ = a b^2 c^2 \]
\[\text{Therefore},\]
\[a * \left( b * c \right) \neq \left( a * b \right) * c\]
Thus, * is not associative on Q.
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