Question:
A binary operation \( * \) on \( \mathbb{Z} \) is defined by:
\[ a * b = 3a + b \]
Determine its properties.
Options:
- (a) Commutative
- (b) Associative
- (c) Not commutative
- (d) Commutative and associative
Solution:
Step 1: Check Commutativity
\[ a * b = 3a + b \] \[ b * a = 3b + a \]
Clearly:
\[ 3a + b \neq 3b + a \quad (\text{in general}) \]
So, operation is not commutative.
—Step 2: Check Associativity
\[ (a * b) * c = (3a + b) * c = 3(3a + b) + c = 9a + 3b + c \]
\[ a * (b * c) = a * (3b + c) = 3a + (3b + c) = 3a + 3b + c \]
Since:
\[ 9a + 3b + c \neq 3a + 3b + c \]
So, operation is not associative.
—Final Answer:
\[ \boxed{\text{(c) Not commutative}} \]
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