Let A and B be two sets defined as follows:
`A = {(x,y) in R xx R : y = sinx}`
`B ={(x,x) : x in R}`

Options

A. `A cap B = {(0,0)}`
B. `A cap B = B`
C. `A cap B = phi`
D. `A DeltaB = A cup B`

Answer

To solve the problem, we need to analyze the sets A and B and find their intersection.

### Step-by-Step Solution:

1. **Define the Sets**:
- Set A is defined as \( A = \{(x, y) \in \mathbb{R} \times \mathbb{R} : y = \sin x\} \).
- Set B is defined as \( B = \{(x, x) : x \in \mathbb{R}\} \).

2. **Understanding Set A**:
- The set A consists of all points where the y-coordinate is equal to the sine of the x-coordinate.
- The sine function oscillates between -1 and 1 for all real numbers \( x \). Thus, the range of \( y \) in set A is \( -1 \leq y \leq 1 \).

3. **Understanding Set B**:
- The set B consists of all points where the x-coordinate is equal to the y-coordinate. This represents the line \( y = x \) in the Cartesian plane.

4. **Finding the Intersection \( A \cap B \)**:
- To find the intersection of sets A and B, we need to find points that satisfy both conditions: \( y = \sin x \) and \( y = x \).
- Setting \( \sin x = x \), we need to find the values of \( x \) for which this equation holds true.

5. **Analyzing the Equation \( \sin x = x \)**:
- The function \( \sin x \) intersects the line \( y = x \) at the origin (0, 0).
- For \( x \) values greater than 0, \( \sin x \) is always less than \( x \) (since the maximum value of \( \sin x \) is 1, and for \( x > 1 \), \( \sin x < x \)).
- For \( x \) values less than 0, \( \sin x \) is always greater than \( x \) (since \( \sin x \) approaches -1 but does not equal \( x \) for negative values).
- Therefore, the only solution to \( \sin x = x \) is \( x = 0 \).

6. **Conclusion**:
- The intersection \( A \cap B \) contains only the point \( (0, 0) \).
- Thus, \( A \cap B = \{(0, 0)\} \).

### Final Answer:
The intersection \( A \cap B = \{(0, 0)\} \).