Prove the following infinite version of the pigeon-hole principle: Suppose that X is an infinite subset, and Y is a finite subset. Then for any function f : X →Y, there is a y E Y such that |ƒ−¹(y)] > ∞. (Hint: By contradiction: Assume that [ƒ−¹(y)| < ∞ for all y ≤ Y and use the fact that Y is finite to obtain a contradiction using the generalized pigeonhole principle)
Prove the following infinite version of the pigeon-hole principle: Suppose that X is an infinite subset, and Y is a finite subset. Then for any function f : X →Y, there is a y E Y such that |ƒ−¹(y)] > ∞. (Hint: By contradiction: Assume that [ƒ−¹(y)| < ∞ for all y ≤ Y and use the fact that Y is finite to obtain a contradiction using the generalized pigeonhole principle)
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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![4. Prove the following infinite version of the pigeon-hole principle: Suppose that X is an infinite
subset, and Y is a finite subset. Then for any function f : X →Y, there is a y E Y such that
|ƒ−¹(y)] > ∞. (Hint: By contradiction: Assume that [ƒ−¹(y)| < ∞ for all y € Y and use the
fact that Y is finite to obtain a contradiction using the generalized pigeonhole principle)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F00e5c5de-298c-41fb-9861-0262471f9c17%2F60841079-aa57-408e-a65c-b8ae6c7f9f3b%2F8ui2xdu_processed.png&w=3840&q=75)
Transcribed Image Text:4. Prove the following infinite version of the pigeon-hole principle: Suppose that X is an infinite
subset, and Y is a finite subset. Then for any function f : X →Y, there is a y E Y such that
|ƒ−¹(y)] > ∞. (Hint: By contradiction: Assume that [ƒ−¹(y)| < ∞ for all y € Y and use the
fact that Y is finite to obtain a contradiction using the generalized pigeonhole principle)
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