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Show that f-(A U B) = f-'(A) U f-'(B) and f-1(A n B) = f-'(A) nf-'(B) for any function f. Is the same true of images (as opposed to inverse images)?

Show that f-(A U B) = f-'(A) U f-'(B) and f-1(A n B) = f-'(A) nf-'(B) for any function f. Is the same true of images (as opposed to inverse images)?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Title: Exploring Properties of Inverse Images in Function Sets** **Introduction:** In this section, we will explore the properties of inverse images for union and intersection sets in the context of functions. This involves analyzing how the inverse function interacts with set operations such as union and intersection. **Problem Statement:** We are tasked to show the following properties for any function \( f \): 1. \( f^{-1}(A \cup B) = f^{-1}(A) \cup f^{-1}(B) \) 2. \( f^{-1}(A \cap B) = f^{-1}(A) \cap f^{-1}(B) \) Additionally, we will consider whether these properties hold true for images, as opposed to inverse images. **Explanation:** - **Inverse Image of a Union:** The first property states that the inverse image of a union of two sets \( A \) and \( B \) is equivalent to the union of their individual inverse images. This means that for any element in the domain of \( f \), it maps to either \( A \), \( B \), or both. - **Inverse Image of an Intersection:** The second property asserts that the inverse image of an intersection of two sets is the intersection of their inverse images. This implies that for an element to be in the inverse image of \( A \cap B \), it needs to map to elements in both sets \( A \) and \( B \). **Discussion:** - **Images and Set Operations:** Unlike inverse images, the image of a union is generally not equal to the union of the images, and similarly for intersection. This is due to the nature of function mapping, where one element from the domain maps to exactly one element in the codomain, but not vice versa. In conclusion, the properties of inverse images behave predictably with unions and intersections, adhering to intuitive notions of set combination. This is crucial for understanding how functions interact with set operations, providing a foundational insight into higher mathematical concepts involving functions and sets.
Transcribed Image Text:**Title: Exploring Properties of Inverse Images in Function Sets** **Introduction:** In this section, we will explore the properties of inverse images for union and intersection sets in the context of functions. This involves analyzing how the inverse function interacts with set operations such as union and intersection. **Problem Statement:** We are tasked to show the following properties for any function \( f \): 1. \( f^{-1}(A \cup B) = f^{-1}(A) \cup f^{-1}(B) \) 2. \( f^{-1}(A \cap B) = f^{-1}(A) \cap f^{-1}(B) \) Additionally, we will consider whether these properties hold true for images, as opposed to inverse images. **Explanation:** - **Inverse Image of a Union:** The first property states that the inverse image of a union of two sets \( A \) and \( B \) is equivalent to the union of their individual inverse images. This means that for any element in the domain of \( f \), it maps to either \( A \), \( B \), or both. - **Inverse Image of an Intersection:** The second property asserts that the inverse image of an intersection of two sets is the intersection of their inverse images. This implies that for an element to be in the inverse image of \( A \cap B \), it needs to map to elements in both sets \( A \) and \( B \). **Discussion:** - **Images and Set Operations:** Unlike inverse images, the image of a union is generally not equal to the union of the images, and similarly for intersection. This is due to the nature of function mapping, where one element from the domain maps to exactly one element in the codomain, but not vice versa. In conclusion, the properties of inverse images behave predictably with unions and intersections, adhering to intuitive notions of set combination. This is crucial for understanding how functions interact with set operations, providing a foundational insight into higher mathematical concepts involving functions and sets.
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