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Question
On Z an operation * is defined by a * b = a2 + b2 for all a, b ∈ Z. The operation * on Z is _______________ .
Options
commutative and associative
associative but not commutative
not associative
not a binary operation
Solution
not associative
Commutativity:
\[\text{ Leta } , b \in Z . \text{ Then }, \]
\[a * b = a^2 + b^2 \]
\[ = b^2 + a^2 \]
\[ = b * a\]
\[\text{ Therefore },\]
\[a * b b * a, \forall a, b \in Z\]
Thus, * is commutative on Z.
Associativity:
\[\text{ Let }a, b, c \in Z\]
\[a * \left( b * c \right) = a * \left( b^2 + c^2 \right)\]
\[ = a^2 + \left( b^2 + c^2 \right)^2 \]
\[ = a^2 + b^4 + c^4 + 2 b^2 c^2 \]
\[\left( a * b \right) * c = \left( a^2 + b^2 \right) * c\]
\[ = \left( a^2 + b^2 \right)^2 + c^2 \]
\[ = a^4 + b^4 + 2 a^2 b^2 + c^2 \]
\[\text{ Therefore },\]
\[a * \left( b * c \right) \neq \left( a * b \right) * c\]
Thus, * is not associative on Z.
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