24 The Pigeonhole Principle

Transcription

1 24 The Pigeonhole Principle Tom Lewis Fall Term 2010 Tom Lewis () 24 The Pigeonhole Principle Fall Term / 9

2 Outline 1 What is the pigeonhole principle 2 Illustrations of the principle 3 Cantor s realms of cardinality Tom Lewis () 24 The Pigeonhole Principle Fall Term / 9

3 What is the pigeonhole principle Where am I going to sleep? Tom Lewis () 24 The Pigeonhole Principle Fall Term / 9

4 What is the pigeonhole principle The pigeonhole principle Suppose we own p pigeons and our coop has h holes. If p h, then the coop is large enough so that pigeons do not have to share holes. However, if p > h, then there are not enough holes to give every pigeon a private room; some pigeons will have to share quarters. Tom Lewis () 24 The Pigeonhole Principle Fall Term / 9

5 What is the pigeonhole principle The pigeonhole principle Suppose we own p pigeons and our coop has h holes. If p h, then the coop is large enough so that pigeons do not have to share holes. However, if p > h, then there are not enough holes to give every pigeon a private room; some pigeons will have to share quarters. Problem Let A and B be finite sets and let f : A B be a function. Recast the pigeonhole principle into the language of functions. Tom Lewis () 24 The Pigeonhole Principle Fall Term / 9

6 Problem Let n N. There there exist positive integers a and b, with a b, such that n a n b is divisible by 10. Tom Lewis () 24 The Pigeonhole Principle Fall Term / 9

7 Problem Let n N. There there exist positive integers a and b, with a b, such that n a n b is divisible by 10. Example Test this out for n = 2 and n = 3. List the first few consecutive powers of each. Does this suggest a proof? Tom Lewis () 24 The Pigeonhole Principle Fall Term / 9

8 Five lattice points Tom Lewis () 24 The Pigeonhole Principle Fall Term / 9

9 Five lattice points Problem Given 5 distinct lattice points in the plane, at least one of the line segments determined by these points has a lattice point as its midpoint. Tom Lewis () 24 The Pigeonhole Principle Fall Term / 9

10 Definition Given a finite sequence S of numbers, a subsequence of S is a list obtained by deleting some of the elements of S. In forming the subsequence, the order of the elements is retained. Tom Lewis () 24 The Pigeonhole Principle Fall Term / 9

11 Definition Given a finite sequence S of numbers, a subsequence of S is a list obtained by deleting some of the elements of S. In forming the subsequence, the order of the elements is retained. Theorem (Erdos-Szerkes) Let n be a positive integer. Every sequence of n distinct integers must contain a monotone subsequence of length n + 1. Tom Lewis () 24 The Pigeonhole Principle Fall Term / 9

12 Definition Given a finite sequence S of numbers, a subsequence of S is a list obtained by deleting some of the elements of S. In forming the subsequence, the order of the elements is retained. Theorem (Erdos-Szerkes) Let n be a positive integer. Every sequence of n distinct integers must contain a monotone subsequence of length n + 1. Example For n = 3, we consider a sequence of 10 numbers: 32, 15, 108, 57, 68, 11, 17, 75, 98, 101 Can we find a subsequence of size 4 that is either increasing or decreasing? Tom Lewis () 24 The Pigeonhole Principle Fall Term / 9

13 Warning! Our next problem is a subtle use of the principle. Tom Lewis () 24 The Pigeonhole Principle Fall Term / 9

14 Warning! Our next problem is a subtle use of the principle. Problem n businessmen meet at a national convention and a sequence of handshakes is exchanged. Show that at least two of the businessmen shake the same number of hands. Tom Lewis () 24 The Pigeonhole Principle Fall Term / 9

15 Cantor s realms of cardinality Definition Let A and B be sets. We say that A and B have the same cardinality provided that there exists a one-to-one and onto function f : A B. Tom Lewis () 24 The Pigeonhole Principle Fall Term / 9

16 Cantor s realms of cardinality Definition Let A and B be sets. We say that A and B have the same cardinality provided that there exists a one-to-one and onto function f : A B. Theorem (Cantor) Let A be a set. If f : A 2 A, then f is not onto. Tom Lewis () 24 The Pigeonhole Principle Fall Term / 9

17 Cantor s realms of cardinality Definition Let A and B be sets. We say that A and B have the same cardinality provided that there exists a one-to-one and onto function f : A B. Theorem (Cantor) Let A be a set. If f : A 2 A, then f is not onto. Remark While this theorem is obvious for finite sets, it is still true for infinite sets. Tom Lewis () 24 The Pigeonhole Principle Fall Term / 9