Question:
On \( \mathbb{Z} \), define:
\[ a * b = a^2 + b^2 \]
Determine its properties.
Options:
- (a) Commutative and associative
- (b) Associative but not commutative
- (c) Not associative
- (d) Not a binary operation
Solution:
Step 1: Check Closure
Since \(a^2 + b^2 \in \mathbb{Z}\), operation is closed ⇒ valid binary operation.
—Step 2: Check Commutativity
\[ a * b = a^2 + b^2 = b^2 + a^2 = b * a \]
So, operation is commutative.
—Step 3: Check Associativity
\[ (a * b) * c = (a^2 + b^2) * c = (a^2 + b^2)^2 + c^2 \]
\[ a * (b * c) = a * (b^2 + c^2) = a^2 + (b^2 + c^2)^2 \]
Clearly:
\[ (a^2 + b^2)^2 + c^2 \neq a^2 + (b^2 + c^2)^2 \]
So, operation is not associative.
—Final Answer:
\[ \boxed{\text{(c) Not associative}} \]