Determine whether or not each of the definition of `'**'` given below gives a binary operation. In the event that `'**'` is not a binary operation, given justification for this:
(i) On `Z^(+)`, define `'**'` by `a**b=a-b`
(ii) On `Z^(+)`, define `'**'` by `a**b=ab`
(iii) On R, define `'**'` by `a**b=ab^(2)`
(iv) On `Z^(+)`, define `'**'` by `a**b=|a-b|`
(v) On Z^(+), define `'**'` by `a**b=a`.

Answer

To determine whether each definition of the operation '**' gives a binary operation, we need to check if the operation is closed within the given set. A binary operation on a set \( S \) is a function that combines two elements from \( S \) to produce another element in \( S \).

Let's analyze each case step by step:

### (i) On \( Z^+ \), define \( a ** b = a - b \)

1. **Choose elements from \( Z^+ \)**: Let \( a = 1 \) and \( b = 2 \).
2. **Calculate \( a ** b \)**: \( 1 - 2 = -1 \).
3. **Check the result**: The result \(-1\) is not in \( Z^+ \) (the set of positive integers).
4. **Conclusion**: Since the operation does not yield an element in \( Z^+ \), \( ** \) is **not a binary operation**.

### (ii) On \( Z^+ \), define \( a ** b = ab \)

1. **Choose elements from \( Z^+ \)**: Let \( a = 2 \) and \( b = 3 \).
2. **Calculate \( a ** b \)**: \( 2 \times 3 = 6 \).
3. **Check the result**: The result \( 6 \) is in \( Z^+ \).
4. **Conclusion**: Since the operation yields an element in \( Z^+ \) for any positive integers \( a \) and \( b \), \( ** \) is a **binary operation**.

### (iii) On \( R \), define \( a ** b = ab^2 \)

1. **Choose elements from \( R \)**: Let \( a = 2 \) and \( b = 3 \).
2. **Calculate \( a ** b \)**: \( 2 \times 3^2 = 2 \times 9 = 18 \).
3. **Check the result**: The result \( 18 \) is in \( R \).
4. **Conclusion**: Since the operation yields an element in \( R \) for any real numbers \( a \) and \( b \), \( ** \) is a **binary operation**.

### (iv) On \( Z^+ \), define \( a ** b = |a - b| \)

1. **Choose elements from \( Z^+ \)**: Let \( a = 3 \) and \( b = 5 \).
2. **Calculate \( a ** b \)**: \( |3 - 5| = | -2 | = 2 \).
3. **Check the result**: The result \( 2 \) is in \( Z^+ \).
4. **Conclusion**: Since the operation yields a positive integer for any positive integers \( a \) and \( b \), \( ** \) is a **binary operation**.

### (v) On \( Z^+ \), define \( a ** b = a \)

1. **Choose elements from \( Z^+ \)**: Let \( a = 4 \) and \( b = 6 \).
2. **Calculate \( a ** b \)**: \( 4 \).
3. **Check the result**: The result \( 4 \) is in \( Z^+ \).
4. **Conclusion**: Since the operation always yields the first element \( a \) (which is in \( Z^+ \)), \( ** \) is a **binary operation**.

### Summary of Results
- (i) Not a binary operation.
- (ii) Binary operation.
- (iii) Binary operation.
- (iv) Binary operation.
- (v) Binary operation.