For each binary operation `'**'` defined below, determine whether `'**'` is commutative and whether `'**'` is associative :
(ii) On Q, define `'**'` by `a**b=ab-1`

Answer

To determine whether the binary operation '**' defined by \( a ** b = ab - 1 \) on the set of rational numbers \( Q \) is commutative and associative, we will analyze both properties step by step.

### Step 1: Check for Commutativity

A binary operation is commutative if for all \( a, b \in Q \), the following holds:
\[ a ** b = b ** a \]

**Calculation:**
1. Calculate \( a ** b \):
\[ a ** b = ab - 1 \]
2. Calculate \( b ** a \):
\[ b ** a = ba - 1 \]
3. Since multiplication of real numbers is commutative, we have \( ab = ba \). Thus:
\[ b ** a = ab - 1 \]
4. Therefore, we find:
\[ a ** b = b ** a \]

**Conclusion for Commutativity:**
Since \( a ** b = b ** a \) holds for all \( a, b \in Q \), the operation '**' is commutative.

### Step 2: Check for Associativity

A binary operation is associative if for all \( a, b, c \in Q \), the following holds:
\[ (a ** b) ** c = a ** (b ** c) \]

**Calculation:**
1. Calculate \( (a ** b) ** c \):
- First, find \( a ** b \):
\[ a ** b = ab - 1 \]
- Now calculate \( (a ** b) ** c \):
\[ (a ** b) ** c = (ab - 1) ** c = (ab - 1)c - 1 = abc - c - 1 \]

2. Calculate \( a ** (b ** c) \):
- First, find \( b ** c \):
\[ b ** c = bc - 1 \]
- Now calculate \( a ** (b ** c) \):
\[ a ** (b ** c) = a ** (bc - 1) = a(bc - 1) - 1 = abc - a - 1 \]

3. Now we compare \( (a ** b) ** c \) and \( a ** (b ** c) \):
- From the calculations:
\[ (a ** b) ** c = abc - c - 1 \]
\[ a ** (b ** c) = abc - a - 1 \]

4. Since \( abc - c - 1 \neq abc - a - 1 \) for general values of \( a, b, c \), we conclude that:
\[ (a ** b) ** c \neq a ** (b ** c) \]

**Conclusion for Associativity:**
Since \( (a ** b) ** c \neq a ** (b ** c) \) for all \( a, b, c \in Q \), the operation '**' is not associative.

### Final Answer:
- The operation '**' is **commutative** but **not associative**.