Dirichlets principle and some applications

Dirichlets Principle and some applications Dirichlets Principle and some applications Thai Nguyen University of education Term Discrete math Dirichlets 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 devote.

Dirichlet's Principle and some applications

Thai Nguyen University of

education

Term: Discrete math

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.

INTRODUCTION

1

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

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

Appication of Dirichlet Principle to other problems

Apply Dirichlet’s principle in proving inequality.

Aproximate a real number

CHAPTER 1

1.1 1.2 1.3 1.4 1.5 1.6 CHAPTER 2

CHAPTER 3

CHAPTER 4

CHAPTER 5

5.1 5.2

2

Chapter 1: Basic

knownledge

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

1.1 Basic Dirichlet's

If k is a positive integer and k+1 or more objects are placed into k boxes, then there is at least one box containing two or more of the objects.

(i) = min{n | n x}.

(ii) x – 1 < x < x + 1.

(iii) – =

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

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

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

equivalent corresponds to an element of B, then there exist at least

two distinct elements of A that correspond to an element of B

1.4 Dirichlet's principle of set

form

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

to other principles.

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

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