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5. Let f: X→Y be a function. (a) Show that A C {(f(A)) for any A E P(X). (b) Show that f(f(B)) C B for any BE P(Y). L4L

5. Let f: X→Y be a function. (a) Show that A C {(f(A)) for any A E P(X). (b) Show that f(f(B)) C B for any BE P(Y). L4L

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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## Mathematical Problems and Proofs ### 4. Function Properties and Relationships Let \( f: X \to Y \) and \( g: Y \to X \) be functions such that \( f \circ g \circ f = f \). #### (a) Prove that \( f \) is surjective if and only if \( f \circ g = \text{id}_Y \). #### (b) Prove that \( f \) is injective if and only if \( g \circ f = \text{id}_X \). #### (c) Prove that if \( f \) is injective, then \( g \) is surjective. #### (d) Prove that if \( f \) is surjective, then \( g \) is injective. ### 5. Analysis of Function \( f: X \to Y \) Let \( f: X \to Y \) be a function. #### (a) Show that \( A \subseteq \tilde{f}(f(A)) \) for any \( A \in \mathcal{P}(X) \). #### (b) Show that \( \tilde{f}(f(B)) \subseteq B \) for any \( B \in \mathcal{P}(Y) \). ### 6. Set Theory and Cardinality Let \( A, B \) be finite sets. Show that if \(|A \setminus B| = |B \setminus A|\), then \(|A| = |B|\).
Transcribed Image Text:## Mathematical Problems and Proofs ### 4. Function Properties and Relationships Let \( f: X \to Y \) and \( g: Y \to X \) be functions such that \( f \circ g \circ f = f \). #### (a) Prove that \( f \) is surjective if and only if \( f \circ g = \text{id}_Y \). #### (b) Prove that \( f \) is injective if and only if \( g \circ f = \text{id}_X \). #### (c) Prove that if \( f \) is injective, then \( g \) is surjective. #### (d) Prove that if \( f \) is surjective, then \( g \) is injective. ### 5. Analysis of Function \( f: X \to Y \) Let \( f: X \to Y \) be a function. #### (a) Show that \( A \subseteq \tilde{f}(f(A)) \) for any \( A \in \mathcal{P}(X) \). #### (b) Show that \( \tilde{f}(f(B)) \subseteq B \) for any \( B \in \mathcal{P}(Y) \). ### 6. Set Theory and Cardinality Let \( A, B \) be finite sets. Show that if \(|A \setminus B| = |B \setminus A|\), then \(|A| = |B|\).
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