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l02_pigeonhole

# l02_pigeonhole - Pigeonhole Principle Reading Ross Ch 1 Sec...

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01/21/2009 CS206 - Intro. to Discrete Structures II 1 Pigeonhole Principle Reading: Ross, Ch 1., Sec. 1. Rosen, Ch 5., Sec. 2 Wednesday, January 28, 2009

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01/21/2009 CS206 - Intro. to Discrete Structures II 2 Objects and Boxes Suppose there are 13 pigeons in a flock and 12 pigeonholes for them to roost. Will all pigeons be able to find their own hole and not share it with another pigeon? No! Proof by contradiction: (adversarial) place one pigeon in each box At most 12 pigeons. Wednesday, January 28, 2009
01/21/2009 CS206 - Intro. to Discrete Structures II Pigeonhole Principle (Dirichlet Drawer Principle) If k is a positive integer and k+1 (or more) objects are placed in k boxes, there must be at least one box with 2 or more objects in it. Consequence: there is no one-to-one function from a set of k to a set of k-1 elements 3 Wednesday, January 28, 2009

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01/21/2009 CS206 - Intro. to Discrete Structures II Examples Among a group of 8 people at least (no fewer than) two must have been born on the same day of a week. If there are 10 benches with 2 seats and 11 students, at least one pair will share a bench. If there are 7 possible grades in a course (A,B+,B,…) how many students should there be in the course so that at least two get the same grade? A: 8. 4 Wednesday, January 28, 2009
01/21/2009 CS206 - Intro. to Discrete Structures II Examples Prove that in any group of three positive integers there are at least two whose sum is even. Let there be two boxes: even & odd. At least two integers have to be in the same box. In either case (two even, two odd) their sum will be even. Prove that in any set of 700 English words, there must be at

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l02_pigeonhole - Pigeonhole Principle Reading Ross Ch 1 Sec...

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