Question

Let \(\mathrm{A}=\left\\{(\mathrm{x}, \mathrm{y}): \mathrm{y}=\mathrm{e}^{\mathrm{x}}, \mathrm{x} \in \mathrm{R}\right\\}, \mathrm{B}=\left\\{(\mathrm{x}, \mathrm{y}): \mathrm{y}=\mathrm{e}^{-\mathrm{x}}, \mathrm{x} \in \mathrm{R}\right\\}\) then (a) \(\mathrm{A} \cap \mathrm{B}=\Phi\) (b) \(\mathrm{A} \cap \mathrm{B} \neq \Phi\) (c) \(A \cup B=R\) (d) \(A \cup B=A\)

Short answer

The correct statement about the relationship between sets A and B is (b): The intersection of A and B is not an empty set (A ∩ B ≠ Φ).

Step-by-step solution

Intersection of A and B (

) First, we need to find the intersection of A and B (A ∩ B). The intersection of A and B consists of all the points (x, y) that belong to both A and B. For a point (x, y) to belong to A, it must satisfy the condition y = e^x. For a point (x, y) to belong to B, it must satisfy the condition y = e^(-x). We will now set e^x = e^(-x) to find if there's any x for which this equation holds true. \(e^x = e^{-x}\) \(e^{2x} = 1\) As we know, \(e^0 = 1\). So, if x = 0, then both exponents are 0, and the equation holds \(e^{2(0)} = 1\) \(e^0 = 1\) Now, let's check for the corresponding y value when x = 0. For A: y = e^x = e^0 = 1 For B: y = e^(-x) = e^(-0) = e^0 = 1 So, (0, 1) is the point that belongs to both A and B, and therefore, A ∩ B ≠ Φ. Hence, option (b) is correct.

Union of A and B (

) Now, let's find the union of sets A and B (A ∪ B). The union of A and B consists of all the points (x, y) that belong to either A or B or both sets. Since both A and B include exponential functions with base e, their union represents all points (x, y) in ℝ where x can be any real number. However, notice that while A represents the set of all points with positive y values (y = e^x, where x ∈ R), B represents the set of all points with positive y values as well (y = e^(-x), where x ∈ R). Therefore, the union of A and B doesn't cover negative y values. Since A ∪ B doesn't represent all real numbers (it doesn't include negative y values), option (c) is incorrect. Now let's check the union of the sets. Since set A contains all positive y values for x ∈ R, and set B also contains all positive y values for x ∈ R, A ∪ B includes all the points that belong to A and all the points that belong to B. However, A and B are not the same sets, so A ∪ B ≠ A. Option (d) is incorrect. #Conclusion#: Based on the results of our analysis: - The correct answer for the intersection is A ∩ B ≠ Φ (option b). - The correct answer for the union is A ∪ B ≠ R and A ∪ B ≠ A (options c and d are incorrect).

Key concepts

Exponential Functions

Exponential functions are mathematical expressions where the variable appears in the exponent. They can be written in the general form as \( f(x) = a \cdot b^x \), where \( a \) is a constant, \( b \) is the base, and \( x \) is the exponent. In this exercise, the base is Euler's number, \( e \), which is approximately 2.718. This special number is widely used in mathematics due to its unique properties, especially in calculus and complex analysis. Understanding exponential growth involves recognizing how rapidly exponential functions increase or decrease. If the base \( b \) is greater than 1, the function represents exponential growth. Conversely, if \( b \,(< 1)\), like \( e^{-x} \), the function represents exponential decay. The graph of \( y = e^x \) is always above the x-axis because exponential functions only yield positive results when the base is a positive real number.Real-world applications of exponential functions are abundant, such as modeling population growth, radioactive decay, and compound interest. Comprehending these functions helps one easily identify their unique behavior and significance in various scientific and financial fields.

Intersection and Union of Sets

Sets are collections of distinct objects or numbers. The intersection and union are two fundamental operations that express relationships between sets.The **intersection** of two sets, denoted as \( A \cap B \), includes all elements that are common to both sets. In this problem, calculating \( A \cap B \) means looking for points \((x, y)\) that satisfy both \( y = e^x \) and \( y = e^{-x} \). Through the solution, setting the expressions equal provided \( e^{x} = e^{-x} \), leading to the solution \( x = 0 \). Thus, the only valid point in the intersection is \((0, 1)\), confirming that \( A \cap B eq \emptyset \) (not empty). The **union** of two sets, expressed as \( A \cup B \), consists of all elements from both sets. For this exercise, it means all points \((x, y)\) that are in either of the sets \( A \) or \( B \). Here, both sets include positive y-values for all real x-values, but neither includes negative y-values, so \( A \cup B eq \mathbb{R} \). While A covers \( (x, e^x) \) and B covers \( (x, e^{-x}) \), their union still surpasses A alone in coverage yet remains distinct from the full set of real numbers.

Real Numbers

Real numbers constitute an essential set of numbers that encompass all possible values along the continuous number line. This includes integers, fractions, and irrational numbers. They play a crucial role in the foundational concepts of calculus and analysis. **Properties of Real Numbers:**

• Completeness: Every non-empty set of real numbers with an upper bound has a least upper bound (or supremum).
• Density: Between any two real numbers lies another real number, which makes the set densely packed along the number line.
• Real Functions: Many functions map real numbers to real numbers, providing breadth for complex calculations and graphing.

In the context of the exercise, the real numbers are where the exponential functions \( y = e^x \) and \( y = e^{-x} \) are defined, meaning they both take all real number values for \( x \). Understanding this range is vital when assessing functions and their intersections or unions, as these operations hinge on knowing the behaviors and coverage of real numbers in these contexts.