b) Suppose \( f: X \to Y \), and \( A, B \subseteq Y \). Prove that(a) \( f^{-1}(A \cup B) = f^{-1}(A) \cup f^{-1}(B) \).(b) \( f^{-1}(A \cap B) = f^{-1}(A) \cap f^{-1}(B) \).

Answered: Image /qna-images/answer/67c48168-8b9f-4212-981b-63d4cc48dd02.jpg

Answered: Suppose f: X→Y, and A, B CY. Prove that…

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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Allergic reactions to poison ivy can be miserable. Plant oils cause the reaction. Researchers at Allergy Institute did a study to determine the effects of washing the oil off within 5 minutes of exposure. A random sample of 1000 people with known allergies to poison ivy participated in the study. Oil from the poison ivy plant was rubbed on a patch of skin. For 500 of the subjects, it was washed off within 5 minutes. For the other 500 subjects, the oil was washed off after 5 minutes. The results are summarized in the following table. Reaction None Mild Strong Column Total Within 5 Minutes After 5 Minutes 391 65 44 500 66 341 93 500 Row Total 457 406 137 1000 Let's use the following notation for the various events: W= washing oil off within 5 minutes, A = washing oil off after 5 minutes, N = no reaction, M = mild reaction, S = strong reaction. Find the following probabilities for a person selected at random from this sample of 1000 subjects. (Use 3 decimal places.)
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Suppose f: X→Y, and A, B CY. Prove that (a) f-¹(AUB) = ƒ−¹(A) U ƒ−¹(B). (b) f-¹(ANB) = f¯¹(A) n f¯¹(B).