Let * be a binary operation on `R` defined by `a*b=a b+1` . Then, * is commutative but not associative associative but not commutative neither commutative nor associative (d) both commutative and associative

Explanation

To determine whether the binary operation defined by \( a * b = ab + 1 \) is commutative and/or associative, we will analyze both properties step by step.

### Step 1: Check for Commutativity
A binary operation is commutative if:
\[ a * b = b * a \quad \text{for all } a, b \in \mathbb{R} \]

**Calculation:**
1. Calculate \( a * b \):
\[ a * b = ab + 1 \]
2. Calculate \( b * a \):
\[ b * a = ba + 1 \]
3. Since multiplication is commutative, \( ab = ba \):
\[ b * a = ab + 1 \]
4. Thus, we have:
\[ a * b = b * a \]

**Conclusion:**
The operation is commutative.

### Step 2: Check for Associativity
A binary operation is associative if:
\[ (a * b) * c = a * (b * c) \quad \text{for all } a, b, c \in \mathbb{R} \]

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

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

3. Now we compare \( (a * b) * c \) and \( a * (b * c) \):
\[ (a * b) * c = abc + c + 1 \]
\[ a * (b * c) = abc + a + 1 \]

**Conclusion:**
Since \( abc + c + 1 \neq abc + a + 1 \) for all \( a, b, c \) (unless \( a = c \)), the operation is not associative.

### Final Conclusion
The binary operation defined by \( a * b = ab + 1 \) is **commutative but not associative**.

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