If the sets A and B are defined are defined as `A={(x,y):y=e^x, x in R},B={(x,y):y=x, x in R}`then

Options

A. `B sub A`
B. `A sub B`
C. `A nn B = phi`
D. `A uu B`

Explanation

To solve the problem, we need to analyze the sets A and B defined as follows:

- Set A: \( A = \{(x, y) : y = e^x, x \in \mathbb{R}\} \)
- Set B: \( B = \{(x, y) : y = x, x \in \mathbb{R}\} \)

### Step-by-Step Solution:

1. **Understanding Set A**:
- The set A represents all points \((x, y)\) where \(y\) is equal to \(e^x\). This is the exponential function which is always positive for all real numbers \(x\). The graph of this function starts from the point \((0, 1)\) and increases rapidly as \(x\) increases.

2. **Understanding Set B**:
- The set B represents all points \((x, y)\) where \(y\) is equal to \(x\). This is a straight line that passes through the origin (0, 0) and has a slope of 1.

3. **Graphing the Functions**:
- To visualize the relationship between the two sets, we can sketch the graphs of both functions:
- The graph of \(y = e^x\) is an upward-curving exponential function that never touches or crosses the x-axis.
- The graph of \(y = x\) is a straight line that passes through the origin and continues indefinitely in both directions.

4. **Finding Points of Intersection**:
- We need to find if there are any points where the two graphs intersect, which means we need to solve the equation:
\[ e^x = x \]
- To analyze this, we can consider the behavior of both functions:
- For \(x < 0\), \(e^x\) is positive and \(x\) is negative, so they cannot intersect.
- For \(x = 0\), \(e^0 = 1\) and \(0 = 0\), so they do not intersect at this point.
- For \(x > 0\), \(e^x\) grows much faster than \(x\), indicating that they will not intersect.

5. **Conclusion**:
- Since there are no points \((x, y)\) that satisfy both conditions simultaneously, we conclude that the intersection of sets A and B is empty:
\[ A \cap B = \emptyset \]

### Final Answer:
The relation between sets A and B is:
\[ A \cap B = \emptyset \]