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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.
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 Table of Content CHAPTER 1.1 1.2 1.3 1.4 1.5 1.6 CHAPTER 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 CHAPTER Appication of Dirichlet’s Principle to arithmetic Application of Dirichlet Principle in the field of combinatorial theorem CHAPTER 5.1 5.2 Appication of Dirichlet Principle to other problems Apply Dirichlet’s principle in proving inequality Aproximate a real number 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 – < x < x + (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 ≥ cages, there exists a cage with at least [rabbits Proof If n rabbits are kept in m ≥ 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] = [] + 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 − 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 ... Dirichlet principle and its application Table of Content CHAPTER 1.1 1.2 1.3 1.4 1.5 1.6 CHAPTER Basic knownledge Basic Dirichlet's Principle The Generalized Dirichlet Principle Extended dirichlet principle. .. 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... Dirichlet's principle of set form Dirichlet's principle of the extended set Application method Application of Dirichlet Principle to compined geology problem CHAPTER CHAPTER Appication of Dirichlet’s Principle