Dirichlets principle and some applications

Thai Nguyen University of
education

Dirichlet's
Principle and
some applications
Term: Discrete math


INTRODUCTION

Dirichlet's principle is a very effective tool
used to prove many profound results of
mathematics. It especially has many
applications in different areas of
mathematics. This principle in many cases
is easy to prove the existence without
giving a specific method, but in fact in
many problems we just need to show the
existence is enough. This thesis is devoted
to presenting the Dirichlet principle and its
application.

1


Table of Content

CHAPTER 1
1.1
1.2

1.3
1.4
1.5
1.6
CHAPTER 2

Basic knownledge
Basic Dirichlet's Principle
The Generalized Dirichlet Principle
Extended dirichlet principle
Dirichlet's principle of set form
Dirichlet's principle of the extended set
Application method
Application of Dirichlet Principle to compined
geology problem

CHAPTER 3
CHAPTER 4

Appication of Dirichlet’s Principle to arithmetic
Application of Dirichlet Principle in the field of
combinatorial theorem

CHAPTER 5
5.1
5.2

Appication of Dirichlet Principle to other
problems
Apply Dirichlet’s principle in proving inequality.

Aproximate a real number
2


Chapter 1: Basic
knownledge


1.1. Basic Dirichlet's
Principle

If k is a positive integer and k+1 or more objects are placed into k boxes,

a. DEFINITION
Let x be a real number. The ceiling
function of x, denoted by , is defined to
be the least integer that is greater than
or equal to x.

b. REMARK
(i) = min{n | n x}.
(ii) x – 1 < x < x + 1.
(iii) – =

P R E S E N T A T I O N

then there is at least one box containing two or more of the objects.


1.2. The

Generalized
Dirichlet
Principle

If N objects are placed into k
boxes, then there is at least
one box containing at least
objects.


1.3. Extended Dirichlet
principle

If n rabbits are kept in m ≥ 2 cages, there exists a cage with at least [rabbits.
Proof .

If n rabbits are kept in m ≥ 2 cages, then there exists a cage with at least [] rabbits,
here the symbol [α] denotes the integer part of the number α.
We prove that the Extended Dirichlet's Principle is as follows: If otherwise every
rabbit cage does not have up to
[] = [ + 1] = [] + 1
rabbits, then the number of rabbits in each cage is smaller or equal to [] rabbits. From
that, it follows that the total number of rabbits does not exceed m[] ≥ n − 1 rabbits.
This makes no sense since there are n rabbits. Therefore, the hypothesis is false.


1.4. Dirichlet's principle of set
form

Let A and B be two non-empty sets with a finite number of elements,

where the number of elements of A is greater than the number of

equivalent corresponds to an element of B, then there exist at least
two distinct elements of A that correspond to an element of B.

P R E S E N T A T I O
O N
N

elements of B. If by some rule, each element of A gives the


1.5 .Dirichlet's principle of the extended set
Suppose A and B are two finite sets, and S(A), S(B) are denoted by the numbers of elements of A and
B respectively. Suppose there is some natural number k that S(A)>k.S(B) and we have a rule that
corresponds each element of A to an element of B. Then there exist at least k+1 elements of A that
correspond to the same element of B.
Note: When k = 1, we immediately have Dirichlet's principle.


1.6. Application method
Dirichlet's principle may seem so simple, but it is a very
powerful tool used to prove many profound results of
mathematics. Dirichlet's principle is also applied to
problems of geometry, which is demonstrated through the
following system of exercises:
To use Dirichlet's principle, we must make a situation
where "rabbit" is locked in a "cage" and satisfy the following
conditions:

+ The number of "rabbits" must be more than the number
of cages.
+ "Rabbits" must be put in all "cages", but it is not
mandatory that every cage has rabbits.
Often the Dirichlet method is applied together with the
counterargument method. In addition, it can also be applied


Some applications
of Dirichlet's
principle


Chapter 2:Application of
Dirichlet Principle to
compined geology
problem


CHAPTER 3:
APPICATION OF
DIRICHLET’S
PRINCIPLE TO
ARITHMETIC


Chapter 4: Application
of Dirichlet Principle in
the field of
combinatorial theorem

Chapter 5:
Appication of
Dirichlet Principle to
other problems


Thank
you for
listening