Question:
Let \( * \) be defined on \( \mathbb{R} \) by:
\[ a * b = ab + 1 \]
Determine its properties.
Options:
- (a) Commutative but not associative
- (b) Associative but not commutative
- (c) Neither commutative nor associative
- (d) Both commutative and associative
Solution:
Step 1: Check Commutativity
\[ a * b = ab + 1 \] \[ b * a = ba + 1 \]
Since \( ab = ba \), we get:
\[ a * b = b * a \]
So, operation is commutative.
—Step 2: Check Associativity
\[ (a * b) * c = (ab + 1) * c \]
\[ = (ab + 1)c + 1 = abc + c + 1 \]
\[ a * (b * c) = a * (bc + 1) \]
\[ = a(bc + 1) + 1 = abc + a + 1 \]
Since:
\[ abc + c + 1 \ne abc + a + 1 \]
So, operation is not associative.
—Final Answer:
\[ \boxed{\text{(a) Commutative but not associative}} \]