Consider the binary operation * : `R xx R rarr R` and o : `R xx R rarr R` defined as a * b = |a-b| and a o b = a, `AA` a, `b in R`. Is o distributive over * ? Justify your answer.

Answer

To determine whether the operation \( o \) is distributive over the operation \( * \), we need to analyze the definitions of these operations and then check the distributive property.

### Step-by-Step Solution:

1. **Define the Operations**:
- The operation \( * \) is defined as:
\[ a * b = |a - b| \]
- The operation \( o \) is defined as:
\[ a \, o \, b = a \]

2. **Understanding Distributive Property**:
- An operation \( o \) is said to be distributive over another operation \( * \) if:
\[ a \, o \, (b * c) = (a \, o \, b) * (a \, o \, c) \]
- We will check if this holds true for our operations.

3. **Calculate Left-Hand Side (LHS)**:
- We need to compute \( a \, o \, (b * c) \):
\[ b * c = |b - c| \]
Therefore,
\[ a \, o \, (b * c) = a \, o \, |b - c| = a \]

4. **Calculate Right-Hand Side (RHS)**:
- Now, we compute \( (a \, o \, b) * (a \, o \, c) \):
\[ a \, o \, b = a \quad \text{and} \quad a \, o \, c = a \]
Thus,
\[ (a \, o \, b) * (a \, o \, c) = a * a = |a - a| = 0 \]

5. **Compare LHS and RHS**:
- From our calculations:
- LHS: \( a \)
- RHS: \( 0 \)
- Since \( a \neq 0 \) for arbitrary \( a \), we conclude:
\[ a \, o \, (b * c) \neq (a \, o \, b) * (a \, o \, c) \]

6. **Conclusion**:
- Since the left-hand side does not equal the right-hand side, we conclude that the operation \( o \) is **not distributive** over the operation \( * \).

### Final Answer:
The operation \( o \) is not distributive over \( * \).