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pigeonholekey2 - McCombs Math 381 The Pigeonhole Principle...

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McCombs Math 381 The Pigeonhole Principle 1 The Pigeonhole Principle: Let A and B be finite sets with a function f : A ! B . (i) If A > B , then f is not one-to-one. (ii) If A < B , then f is not onto. Basic Idea: If n objects are put into k boxes, where n > k > 0 , then at least one box must hold more than one object. Generalized Pigeonhole Principle: For integers N > k > 0 , if N objects are put into k boxes, then at least one box must contain at least N k ! " " # $ $ objects, where N k ! " " # $ $ = the smallest integer that is greater than or equal to N k . In other words, if N = m ! k + 1 objects are put into k boxes, where k > 0, m > 0 , then at least one box must hold at least m + 1 objects. Basic Strategy: 1. Identify the objects to be placed in boxes. These are the things that you’d like several of to share some special property. 2. Set up the boxes. You want to do this so that when you get two or more objects in the same box, they share the desired property. Important Note: Make sure that your setup yields fewer boxes than objects to be placed. 3. Devise a rule for assigning objects to boxes in such a way that “enough” objects occupy the same box.
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