The binary operation * defined on `N` by a*b=`a+b+a b` for all `a ,\ b in N` is (a) commutative only (b) associative only (c) commutative and associative both (d) none of these

Options

A. commutative and associative both
B. associative only
C. commutative and
associative both
D. None of these

Explanation

To determine whether the binary operation \( * \) defined on \( \mathbb{N} \) by \( a * b = a + b + ab \) is commutative, associative, both, or neither, we will analyze the properties step by step.

### Step 1: Check for Commutativity

A binary operation is commutative if \( a * b = b * a \) for all \( a, b \in \mathbb{N} \).

**Left Hand Side (LHS):**
\[ a * b = a + b + ab \]

**Right Hand Side (RHS):**
\[ b * a = b + a + ba \]

Since multiplication is commutative, \( ab = ba \). Therefore, we can rewrite the RHS as:
\[ b * a = b + a + ab \]

Now we see that:
\[ a * b = a + b + ab = b + a + ab = b * a \]

Thus, \( a * b = b * a \), which means the operation is **commutative**.

### Step 2: Check for Associativity

A binary operation is associative if \( (a * b) * c = a * (b * c) \) for all \( a, b, c \in \mathbb{N} \).

**Left Hand Side (LHS):**
First, we calculate \( a * b \):
\[ a * b = a + b + ab \]
Now, we need to compute \( (a * b) * c \):
\[ (a * b) * c = (a + b + ab) * c = (a + b + ab) + c + (a + b + ab)c \]
Expanding this gives:
\[ = a + b + ab + c + ac + bc + abc \]

**Right Hand Side (RHS):**
Now we calculate \( b * c \):
\[ b * c = b + c + bc \]
Then we compute \( a * (b * c) \):
\[ a * (b * c) = a * (b + c + bc) = a + (b + c + bc) + a(b + c + bc) \]
Expanding this gives:
\[ = a + b + c + bc + ab + ac + abc \]

### Step 3: Compare LHS and RHS

Now we compare the two results:
- LHS: \( a + b + ab + c + ac + bc + abc \)
- RHS: \( a + b + c + bc + ab + ac + abc \)

Both expressions are equal, thus:
\[ (a * b) * c = a * (b * c) \]

This means the operation is **associative**.

### Conclusion

Since the operation \( * \) is both commutative and associative, the correct answer is:
**(c) commutative and associative both.**

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