For each binary operation `'**'` defined below, determine whether `'**'` is commutative and whether `'**'` is associative :
(vii) On `R-{-1}`, define `'**'` by `a**b=(a)/(b+1)`

Answer

To determine whether the binary operation \( ** \) defined by \( a ** b = \frac{a}{b + 1} \) on the set \( R - \{-1\} \) 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 \( a \) and \( b \):
\[ a ** b = b ** a \]
Substituting the operation definition:
\[ a ** b = \frac{a}{b + 1} \]
\[ b ** a = \frac{b}{a + 1} \]
Now we need to check if:
\[ \frac{a}{b + 1} = \frac{b}{a + 1} \]
Cross-multiplying gives:
\[ a(a + 1) = b(b + 1) \]
This simplifies to:
\[ a^2 + a = b^2 + b \]
This equation does not hold for all \( a \) and \( b \). For example, if we take \( a = 1 \) and \( b = 2 \):
\[ 1^2 + 1 = 2 \quad \text{and} \quad 2^2 + 2 = 6 \]
Since \( 2 \neq 6 \), we conclude that the operation is not commutative.

### Step 2: Check for Associativity
A binary operation is associative if for all \( a, b, c \):
\[ (a ** b) ** c = a ** (b ** c) \]
Calculating the left-hand side:
\[ (a ** b) ** c = \left(\frac{a}{b + 1}\right) ** c = \frac{\frac{a}{b + 1}}{c + 1} = \frac{a}{(b + 1)(c + 1)} \]
Now calculating the right-hand side:
\[ b ** c = \frac{b}{c + 1} \]
Then,
\[ a ** (b ** c) = a ** \left(\frac{b}{c + 1}\right) = \frac{a}{\frac{b}{c + 1} + 1} = \frac{a}{\frac{b + (c + 1)}{c + 1}} = \frac{a(c + 1)}{b + c + 1} \]
Now we need to check if:
\[ \frac{a}{(b + 1)(c + 1)} = \frac{a(c + 1)}{b + c + 1} \]
Cross-multiplying gives:
\[ a(b + c + 1) = a(c + 1)(b + 1) \]
Assuming \( a \neq 0 \) (since \( a \) can be any real number except for \( -1 \)), we can divide both sides by \( a \):
\[ b + c + 1 = (c + 1)(b + 1) \]
Expanding the right-hand side:
\[ b + c + 1 = bc + b + c + 1 \]
This simplifies to:
\[ 0 = bc \]
This is not true for all \( b \) and \( c \). For example, if \( b = 1 \) and \( c = 1 \), then \( 1 \cdot 1 = 1 \), which is not zero. Thus, the operation is not associative.

### Conclusion
The binary operation \( ** \) defined by \( a ** b = \frac{a}{b + 1} \) is neither commutative nor associative.

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