Question

For the binary operation defined on \( \mathbb{R} - \{1\} \) such that: \[ a b = \frac{a}{b + 1} \] which of the following is true?

• A. Not associative
• B. Commutative
• C. Not commutative
• D. Both (A) and (B)

Hint:

Commutative operations satisfy \( a b = b a \), while associative operations satisfy \( (a b) c = a (b c) \).

Answer 1

The Correct Option is D

Step 1: Checking commutativity.
For commutativity, \( a b = b a \): \[ \frac{a}{b + 1} = \frac{b}{a + 1} \] Since this holds, the operation is commutative.
Step 2: Checking associativity.
For associativity, \( (a b) c = a (b c) \), which does not hold, so it is not associative.