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

Answer 1

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.