if set A and B 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 subB`
C. `A nn B=phi`
D. `Acup B =A`

Answer

To solve the problem, we need to analyze the two 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 1: Understand the Graphs of the Functions
1. **Graph of Set A**: The equation \( y = e^x \) represents an exponential function. The graph of this function is always above the x-axis (since \( e^x > 0 \) for all \( x \)), and it approaches the x-axis as \( x \) approaches negative infinity. As \( x \) increases, \( y \) increases rapidly.

2. **Graph of Set B**: The equation \( y = x \) represents a straight line that passes through the origin (0,0) with a slope of 1. This line extends infinitely in both directions and intersects the x-axis and y-axis at the origin.

### Step 2: Analyze the Intersection of Sets A and B
To find the intersection \( A \cap B \), we need to determine if there are any points \((x, y)\) that satisfy both equations simultaneously.

- For a point to be in both sets, it must satisfy both \( y = e^x \) and \( y = x \).
- Setting these equal gives us the equation \( e^x = x \).

### Step 3: Solve the Equation \( e^x = x \)
To analyze the equation \( e^x = x \):
- Consider the function \( f(x) = e^x - x \).
- The function \( f(x) \) is continuous and differentiable.
- At \( x = 0 \), \( f(0) = e^0 - 0 = 1 > 0 \).
- As \( x \to -\infty \), \( f(x) \to \infty \) (since \( e^x \to 0 \) and \( -x \to \infty \)).
- As \( x \to \infty \), \( f(x) \to \infty \) (since \( e^x \) grows faster than \( x \)).
- The derivative \( f'(x) = e^x - 1 \) is zero at \( x = 0 \) and positive for \( x > 0 \), indicating that \( f(x) \) has a minimum at \( x = 0 \).

### Step 4: Determine if There are Solutions
- Since \( f(0) > 0 \) and \( f(x) \) approaches infinity in both directions, we conclude that there are no points where \( e^x = x \).
- Therefore, the intersection \( A \cap B = \emptyset \) (the empty set).

### Step 5: Analyze the Union of Sets A and B
- The union \( A \cup B \) will contain all points from both sets.
- Since \( A \) consists of points where \( y > 0 \) (the exponential curve) and \( B \) consists of points where \( y \) can be any real number (the line), the union will include all points from both sets without any overlap.

### Conclusion
- The intersection of sets A and B is empty: \( A \cap B = \emptyset \).
- The union of sets A and B includes all points from both sets.