Let $A=\{(x,y)|y=e^{x},x\in R\}$, $B=\{(x,y)|y=x,x\in R\}$, then :

Question

Accepted Answer

To determine why Option 3 is the correct answer, let's analyze the sets $ A $ and $ B $ and their intersection.

1. **Set $ A $:** This set is defined as $ A = \{(x, y) \mid y = e^x, x \in \mathbb{R}\} $. This means that for every real number $ x $, the corresponding $ y $ value is $ e^x $. The graph of this set is the exponential curve $ y = e^x $, which is always positive for all real $ x $.

2. **Set $ B $:** This set is defined as $ B = \{(x, y) \mid y = x, x \in \mathbb{R}\} $. This means that for every real number $ x $, the corresponding $ y $ value is equal to $ x $. The graph of this set is the line $ y = x $, which passes through the origin and extends infinitely in both directions.

Now, let's analyze the intersection $ A \cap B $:

- For a point $(x, y)$ to be in both $ A $ and $ B $, it must satisfy both $ y = e^x $ and $ y = x $. Therefore, we need to solve the equation $ e^x = x $.

- The function $ y = e^x $ is always positive, while the function $ y = x $ can be negative, zero, or positive. The exponential function $ e^x $ grows faster than the linear function $ y = x $ and is only equal to $ x $ at specific points.

- By analyzing the functions, we find that $ e^x = x $ has no real solutions. The exponential function $ e^x $ is greater than $ x $ for all $ x $ except for a very small interval around $ x = 0 $, but even there, they do not intersect. Therefore, there are no points $(x, y)$ that satisfy both equations simultaneously.

Thus, the intersection $ A \cap B $ is empty, which means $ A \cap B = \phi $. This justifies why Option 3 is correct.

Now, let's explain why the other options are incorrect:

- **Option 1: $ B \subset A $**: This would imply that every point on the line $ y = x $ is also on the curve $ y = e^x $. However, as shown, the line and the curve do not intersect at any point, so $ B $ cannot be a subset of $ A $.

- **Option 2: $ A \subset B $**: This would imply that every point on the curve $ y = e^x $ is also on the line $ y = x $. Since the exponential curve does not intersect the line, $ A $ cannot be a subset of $ B $.

- **Option 4: $ A \cup B = A $**: This would imply that the union of the two sets is equal to $ A $, meaning every point on $ B $ is also on $ A $. Since they do not intersect, the union cannot be equal to $ A $ alone.

In conclusion, the correct answer is Option 3, $ A \cap B = \phi $, because the sets $ A $ and $ B $ do not share any common points.