Title page for etd-0625109-115205


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URN etd-0625109-115205
Author Wei-cheng Hung
Author's Email Address No Public.
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Department Applied Mathematics
Year 2008
Semester 2
Degree Master
Type of Document
Language zh-TW.Big5 Chinese
Title Inclusion-exclusion and pigeonhole principles
Date of Defense 2009-06-05
Page Count 116
Keyword
  • pigeonhole principle
  • Ramsey theorem
  • derangement
  • inclusion-exclusion principle
  • consecutive permutation
  • Euler’s phi function
  • complete graph
  • combinations with repetition
  • onto function
  • rook polynomial
  • nonnegative integer solutions
  • Abstract In this paper, we will review two fundamental counting methods: inclusionexclusion and pigeonhole principles. The inclusion-exclusion principle considers
    the elements of the sets satisfied some conditions, and avoids repeat counting by disjoint sets. We also use the inclusion-exclusion principle to solve the problems of Euler phi function and the number of onto functions in number theory, and derangement and the number of nonnegative integer solutions of equations in combinatorics. We derive the closed-form formula to those problems. For the forbidden positions problems, we use the rook polynomials to simplify the counting process. We also show the form of the inclusion-exclusion principle in probability, and use it to solve some probability problems.
    The pigeonhole principle is an easy concept. We can establish some sets and use the pigeonhole principle to discuss the extreme value about the number of
    elements. Choose the pigeons and pigeonholes, properly, and solve problems by the concept of the pigeonhole principle. We also introduce the Ramsey theorem which is an important application of the pigeonhole principle. This theorem provides a method to solve problems by complete graph. Finally, we give some contest problems about the inclusion-exclusion and pigeonhole principles to show how those principles are used.
    Advisory Committee
  • Mong-Na Lo Huang - chair
  • Mei-Hui Guo - co-chair
  • Fu-Chuen Chang - advisor
  • Files
  • etd-0625109-115205.pdf
  • indicate not accessible
    Date of Submission 2009-06-25

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