Equivalent Structures on Sets: Equivalence Classes, Partitions and Fiber Structures of Functions
Equivalent Structures on Sets: Equivalence Classes, Partitions and Fiber Structures of Functions
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集合上的等价结构：函数的等价类、划分和纤维结构
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发表时间：
2006
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通讯作者：
M. Hamdan
中科院分区：
文献类型：
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作者：
M. Hamdan
本研究报告了如何引导学生在作为划分的集合、由等价关系确定的等价类集合和该集合(即函数范围内b的所有集合{b}的原象的集合)上的函数的纤维结构之间建立有意义的联系。在这篇文章中，我首先在APOS理论的意义下，对等价关系和函数的概念在它们在集合上确定的结构的上下文中进行了初始遗传分解。这种基因分解主要是基于我自己的数学知识以及我对学生学习过程的观察。基于这一分析，我随后提出了激发基因分解中所描述的精神活动的教学程序。最后，我给出了对处于不同学习阶段的学生的非正式访谈的经验数据。我的目标是引导学生意识到上述结构之间的密切概念对应和联系。一个刻画这种联系的定理如下：集合A上的关系R是等价关系当且仅当存在定义在A上的函数f，使得通过R相关的元素(并且只有那些元素)在f下具有相同的映象。
This study reports on how students can be led to make meaningful connections between such structures on a set as a partition, the set of equivalence classes determined by an equivalence relation and the fiber structure of a function on that set (i.e., the set of preimages of all sets {b} for b in the range of the function). In this paper, I first present an initial genetic decomposition, in the sense of APOS theory, for the concepts of equivalence relation and function in the context of the structures that they determine on a set. This genetic decomposition is primarily based on my own mathematical knowledge as well as on my observations of students’ learning processes. Based on this analysis, I then suggest instructional procedures that motivate the mental activities described in the genetic decomposition. I finally present empirical data from informal interviews with students at different stages of learning. My goal was to guide students to become aware of the close conceptual correspondence and connections among the aforementioned structures. One theorem that captures such connections is the following: a relation R on a set A is an equivalence relation if and only if there exists a function f defined on A such that elements related via R (and only those) have the same image under f.