Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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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)
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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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