Question

Check the commutativity and associativity of the following binary operation '*' on Q defined by a * b = ab² for all a, b ∈ Q ?

Answer 1

 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.