等价关系的应用数学毕业论文范文精品

本科生毕业论文（设计）

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题目

浅谈等价关系在大学数学一些课程中的应用

摘要

等价关系作为集合元素之间的一种特殊二元关系，在大学数学多门课程中均有广泛应用，例如数学分析，高等代数，近世代数，离散数学，点集拓扑等基础课程和专业核心课程．本文首先从等价关系的两种定义出发，通过等价关系的不同定义其在高等代数中的矩阵合同、相似概念；近世代数中的陪集、商群概念；离散数学中的等值式；图论及点集拓扑中的连通关系、商空间等概念，并讨论这些概念在一些课程中的作用．其次，讨论等价关系在高等数学的求解极限中的应用．最后，本文讨论了等价关系在大学课程之外的应用拓展．

关键词：等价关系；相似；陪集；商群；商空间

ABSTRACT

As a special mutual relation within elements of a set, equivalence relation play an important and wide role in the university mathematics courses, such as mathematical analysis, advanced algebra, modern algebra, discrete mathematics, point set topology and other basic curriculum and the professional core courses. Firstly, from the two definition of equivalence relation, this paper define the concepts of matrix similar in Higher Algebra, the conset quotient groups of modern algebra, equivalent type of the discrete mathematics,connected relation, quotient space concepts of graph theory and topology through different equivalence classes and discuss application of these concepts in those courses. Secondly,this paper is using the equivalent relation to solving limit of higher mathematics. Finally, this paper discusses the application development of equivalent relation which is outside of university courses.

Key words: equivalence relation ; similar ; conset ; topology of connected relation ; quotient space

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摘要............................................................................................................ I ABSTRACT . .................................................................................................... II 1 引言 ............................................................................................................... 1 2 基本概念 ....................................................................................................... 2 3 等价关系与集合分类间的关系 ................................................................... 4

3.1 由集合分类唯一确定一等价关系 . ............................................................ 4

3.2 等价关系唯一确定一集合分类 . ................................................................ 4

3.3 简单的应用 . ................................................................................................ 5 4 等价关系在几门课程中的应用 ................................................................... 6

4.1 数学分析中的等价关系 . ............................................................................ 6

4.2 高等代数中的等价关系 . ............................................................................ 9

4.2.1

4.2.2

4.2.3 初等变换 . .............................................................................. 9 矩阵的相似 . .......................................................................... 9 矩阵的合同 . ........................................................................ 10

4.3 等价关系在离散数学中的引出的新概念 . .............................................. 11

4.4 等价关系在近世代数中的引出的新概念 . .............................................. 13

4.4.1

4.4.2 陪集 . .................................................................................... 13 商群 . .................................................................................... 14

4.5 等价关系在点集拓扑中的引出的新概念 . .............................................. 15

4.5.1

商空间 . ................................................................................ 15