反证法解题方法及应用研究(第五稿)

反证法解题方法及应用研究 摘 要 反证法是初等数学解题方法中极其重要的方法之一，特别是当一些直接证明无法入手时，使用反证法证明将会化难为易，所谓“正难则反”便这种方法．[1]反证法主要是运用逆向思维的逻辑来解题，先假设结论的反面成立，再由假设出发，根据已有的定义、公理、定理、条件，使推导得出的结果与原命题的已知条件相矛盾，从而否定假设，达到肯定原命题正确的一种方法．并且利用反证法解题可以提高学生的逻辑思维能力，因此反证法在初等数学解题中得到了广泛的应用．本文主要从反证法的概念及步骤、如何做出正确反设及矛盾推导、论证形式及逻辑原理、反证法适用范围、适用反证法的命题及举例上作了大量论述，并总结出了一套提升反证法解题能力的方法．因此，旨在通过本文对反证法的研究，从而对培养学生的逻辑思维能力和解题技巧有所帮助．

关键词：反证法；逻辑思维；解题技巧；应用；能力提升 Reduction to absurdity problem solving method and application research Abstract: Reduction to absurdity is one of the extremely important method in the elementary mathematics problem-solving method, especially when some directly prove unable to start with, using the reduction will be hard, so-called "is difficult," this kind of method. [1] the reduction to absurdity is mainly using reverse thinking logic to problem solving, the reverse of the first hypothesis conclusion was established, by assumption, again according to the existing definitions, axioms, theorems, conditions, the derived results with the original proposition of the known conditions, thus negative assumptions, to sure the original proposition right a way. And the reduction to absurdity problem solving can be used to improve the students' logical thinking ability, so the reduction to absurdity in elementary mathematics problem-solving has been widely used. This article mainly from the concept and steps of reduction to absurdity, how to make the right inverse derivation set and contradiction, the text argument forms and logical principle, the applicable scope, applicable proposition of reduction to absurdity and made a lot of paper, for example, and summarizes a set of method to improve the reduction to absurdity problem solving skills. Therefore, through the research of reduction to absurdity, to cultivate students' logical thinking ability and problem solving skills.

Key words: reduction to absurdity; Logical thinking; The problem solving skills; Application; Ability to ascend 目 录 1 引言 ................................................................. 1 2 文献综述 ............................................................. 1 2.1 国内外研究现状 ....................................................... 1 2.2 国内外研究现状评价 ................................................... 2 2.3 提出问题 ............................................................. 2 3 反证法的概念及一般步骤 ............................................... 2 3.1 生活中的反证法 ....................................................... 2 3.2 反证法的概念 ......................................................... 3 3.3 反证法的一般步骤 ..................................................... 4 4 反证法如何做出正确反设及矛盾推导 ..................................... 4 4.1 如何做出正确反设 ..................................................... 4 4.2 如何否定命题的结论 ................................................... 6 4.3 如何正确归谬 ......................................................... 7 5 反证法的论证形式及逻辑原理 ........................................... 8 5.1 反证法的论证形式 ..................................................... 8 5.2 反证法的逻辑原理 ..................................................... 8 6 反证法适用范围 ....................................................... 9 6.1 反证法在代数证题中的应用 ............................................. 9 6.2 反证法在三角函数证题中的应用 ........................................ 10 6.3 反证法在平面几何证题中的应用 ........................................ 11 6.4 反证法在立体几何证题中的应用 ........................................ 11 6.5 反证法在解析几何证题中的应用 ........................................ 12 7 适用反证法的命题举例及能力提升 ...................................... 13 7.1 “否定性”命题 ...................................................... 13 7.2 某些涉及无理数的命题 ................................................ 14 7.3 “无限性”命题 ...................................................... 15 7.4 “判断性”命题 ...................................................... 16 7.5 “起始性”命题 ...................................................... 17 7.6 “至多”与“至少”命题 .............................................. 18 7.7 某些命题的“逆命题” ................................................ 19 7.8 “存在性”与“唯一性”命题 .......................................... 20 7.9 “都是”与“存在一个不是”的命题 .................................... 21 7.10 “都不是”与“存在一个是”的命题 .................................... 21 7.11 如何提升反证法解题能力 ............................................. 22 8 结论 ................................................................ 23 8.1 主要发现 ............................................................ 23 8.2 启示 ................................................................ 23 8.3 局限性 .............................................................. 23 8.4 努力方向 ............................................................ 24 参考文献 ............................................................. 25