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or Falsy? : f¹(AUB) = f'(A) f¹(B) 170 Тимо 1 Faise

or Falsy? : f¹(AUB) = f'(A) f¹(B) 170 Тимо 1 Faise

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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**Question:** True or False? \( f^{-1}(A \cup B) = f^{-1}(A) \cap f^{-1}(B) \) **Answer:** False This handwritten note explores the concept of inverse images in set theory, specifically asking whether the inverse image of the union of two sets (\(A\) and \(B\)) is equal to the intersection of their inverse images under a function \(f\). The answer given is "False," indicating that this equality generally does not hold. **Explanation:** The inverse image of a union, \(f^{-1}(A \cup B)\), is the set of all elements in the domain of \(f\) that map to either \(A\) or \(B\) in the codomain. However, the intersection of inverse images, \(f^{-1}(A) \cap f^{-1}(B)\), consists of elements in the domain that map to elements common to both \(A\) and \(B\). Therefore, these two sets are generally different. There are no graphs or diagrams in the image.
Transcribed Image Text:**Question:** True or False? \( f^{-1}(A \cup B) = f^{-1}(A) \cap f^{-1}(B) \) **Answer:** False This handwritten note explores the concept of inverse images in set theory, specifically asking whether the inverse image of the union of two sets (\(A\) and \(B\)) is equal to the intersection of their inverse images under a function \(f\). The answer given is "False," indicating that this equality generally does not hold. **Explanation:** The inverse image of a union, \(f^{-1}(A \cup B)\), is the set of all elements in the domain of \(f\) that map to either \(A\) or \(B\) in the codomain. However, the intersection of inverse images, \(f^{-1}(A) \cap f^{-1}(B)\), consists of elements in the domain that map to elements common to both \(A\) and \(B\). Therefore, these two sets are generally different. There are no graphs or diagrams in the image.
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