Which of the following is not a binary operation on the indicated set?

Options

A. On `Z^(+),**` defined by `a**b=a-b`
B. On `Z^(+),**` defined by `a**b=ab`
C. On `R,**` defined by `a**b=ab^(2)`
D. None of above

Answer

To determine which of the given operations is not a binary operation on the indicated set, we need to understand the definition of a binary operation. A binary operation on a set is a rule that combines two elements from that set to produce another element from the same set.

### Step-by-Step Solution:

1. **Understanding Binary Operations**:
A binary operation on a set \( S \) is a function \( *: S \times S \rightarrow S \). This means that for any two elements \( a, b \in S \), the result of the operation \( a * b \) must also be an element of \( S \).

2. **Analyzing Each Option**:
We will check each operation to see if it satisfies the definition of a binary operation.

- **Option A**: Defined as \( A * B = A - B \) on positive integers \( Z^+ \).
- Let’s take \( A = 2 \) and \( B = 4 \).
- Then, \( A * B = 2 - 4 = -2 \).
- Since \(-2\) is not a positive integer, this operation does not produce an output in the set \( Z^+ \). Therefore, this is **not a binary operation**.

- **Option B**: Defined as \( A * B = A \cdot B \) on positive integers \( Z^+ \).
- For any \( A, B \in Z^+ \), \( A * B = A \cdot B \) will always yield a positive integer since the product of two positive integers is always positive.
- Thus, this operation is a binary operation.

- **Option C**: Defined as \( A * B = A \cdot B^2 \) on real numbers \( R \).
- For any \( A, B \in R \), \( A * B = A \cdot B^2 \).
- Since \( B^2 \) is always non-negative (it can be zero or positive), the product \( A \cdot B^2 \) will also be a real number.
- Therefore, this operation is a binary operation.

3. **Conclusion**:
Since option A does not satisfy the condition of being a binary operation (as it produces a result outside the set of positive integers), the answer is:
\[ \text{Option A is not a binary operation.} \]