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