§24 The Pigeonhole Principle
Tom Lewis
Fall Term 2010
Tom Lewis ()
§24 The Pigeonhole Principle
Fall Term 2010
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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 2010
2/9
What is the pigeonhole principle
Where am I going to sleep?
Tom Lewis ()
§24 The Pigeonhole Principle
Fall Term 2010
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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 2010
4/9
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 2010
4/9
Illustrations of the principle
Problem
Let n ∈ N. There there exist positive integers a and b, with a 6= b, such
that na − nb is divisible by 10.
Tom Lewis ()
§24 The Pigeonhole Principle
Fall Term 2010
5/9
Illustrations of the principle
Problem
Let n ∈ N. There there exist positive integers a and b, with a 6= b, such
that na − nb 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 2010
5/9
Illustrations of the principle
Five lattice points
Tom Lewis ()
§24 The Pigeonhole Principle
Fall Term 2010
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Illustrations of the principle
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 2010
6/9
Illustrations of the principle
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 2010
7/9
Illustrations of the principle
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 n2 + 1 distinct integers
must contain a monotone subsequence of length n + 1.
Tom Lewis ()
§24 The Pigeonhole Principle
Fall Term 2010
7/9
Illustrations of the principle
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 n2 + 1 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 2010
7/9
Illustrations of the principle
Warning!
Our next problem is a subtle use of the principle.
Tom Lewis ()
§24 The Pigeonhole Principle
Fall Term 2010
8/9
Illustrations of the principle
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 2010
8/9
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 2010
9/9
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 → 2A , then f is not onto.
Tom Lewis ()
§24 The Pigeonhole Principle
Fall Term 2010
9/9
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 → 2A , 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 2010
9/9