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Suppose that f: X → Y and let AC X and BCY. a counterexample: counterexample: a (a) Prove or give (b) Prove or give (c) Prove or give a counterexample: (d) Prove or give a counterexample: ƒ−¹(ƒ(A)) ≤ A B ≤ ƒ(ƒ−¹(B)) f(AUf-¹(B)) = f(A) U B. ƒ−¹(ƒ(A)^B) = An ƒ−¹(B)

Suppose that f: X → Y and let AC X and BCY. a counterexample: counterexample: a (a) Prove or give (b) Prove or give (c) Prove or give a counterexample: (d) Prove or give a counterexample: ƒ−¹(ƒ(A)) ≤ A B ≤ ƒ(ƒ−¹(B)) f(AUf-¹(B)) = f(A) U B. ƒ−¹(ƒ(A)^B) = An ƒ−¹(B)

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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Suppose that \( f: X \to Y \) and let \( A \subseteq X \) and \( B \subseteq Y \). (a) Prove or give a counterexample: \( f^{-1}(f(A)) \subseteq A \) (b) Prove or give a counterexample: \( B \subseteq f(f^{-1}(B)) \) (c) Prove or give a counterexample: \( f(A \cup f^{-1}(B)) = f(A) \cup B \) (d) Prove or give a counterexample: \( f^{-1}(f(A) \cap B) = A \cap f^{-1}(B) \)
Transcribed Image Text:Suppose that \( f: X \to Y \) and let \( A \subseteq X \) and \( B \subseteq Y \). (a) Prove or give a counterexample: \( f^{-1}(f(A)) \subseteq A \) (b) Prove or give a counterexample: \( B \subseteq f(f^{-1}(B)) \) (c) Prove or give a counterexample: \( f(A \cup f^{-1}(B)) = f(A) \cup B \) (d) Prove or give a counterexample: \( f^{-1}(f(A) \cap B) = A \cap f^{-1}(B) \)
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