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所在平台: Udemy |
课程主页: https://www.udemy.com/course/applied-mathematics-mathematical-reasoning/
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**应用数学 - 数学推理 课程总结** 本课程《应用数学 - 数学推理》旨在帮助学员掌握数学推理的基础概念和方法。 **核心内容概览：** 1. **数学上可接受的陈述：** 学习识别一个陈述是否为真或为假，即是否为数学上可接受的陈述。 2. **连接词的理解与应用：** * 深入理解和熟练运用“当且仅当（充分必要条件）”、“蕴含”、“与/或”、“被蕴含”、“与”、“或”、“存在”等逻辑连接词。 * 通过丰富的现实生活及数学例子，巩固这些连接词的使用。 * **“如果... 那么...”的多种表述：** 学习将“如果 p，那么 q”的陈述改写为： * p 蕴含 q (p ⇒ q) * p 是 q 的充分条件 * q 是 p 的必要条件 * p 仅当 q * 非 q 蕴含 非 p (∼q ⇒ ∼p) * **逆命题与否命题：** 理解陈述 p ⇒ q 的逆命题 (q ⇒ p) 和否命题 (∼p)。 * **逆否命题：** 掌握陈述 p ⇒ q 的否命题的逆命题，即逆否命题（∼q ⇒ ∼p）。 * **“当且仅当”：** 理解 p ⇒ q 及其逆命题 q ⇒ p 结合起来，便构成“p 当且仅当 q”的陈述。 3. **陈述的有效性验证：** 学习多种方法来检验包含连接词的陈述的有效性： * **直接证明法** * **逆否证法** * **反证法（归谬法）** * **构造反例法** **总结要点：** * 数学上可接受的陈述是明确为真或为假的句子。 * 理解并区分“否定”、“复合命题”及其“分量命题”。 * 掌握“与”，“或”，“存在”，“任意”在复合命题中的作用。 * 深刻理解“蕴含”、“仅当”、“当且仅当”的含义。 * 熟练运用直接证明、逆否证、反证和构造反例等方法来验证陈述的真伪。 通过本课程的学习，学员将能更自信地进行数学论证和逻辑分析。
Mathematical ReasoningMathematically acceptable statementsConnecting words/ phrases - consolidating the understanding of "if and only if (necessary and sufficient) condition", "implies", "and/or", "implied by", "and", "or", "there exists" and their use through variety of examples related to real life and MathematicsValidating the statements involving the connecting words difference between contradiction, converse and contrapositiveSUMMARY1. A mathematically acceptable statement is a sentence which is either true or false. 2. Explained the terms: - Negation of a statement p: If p denote a statement, then the negation of p is denoted by ∼p. - Compound statements and their related component statements: A statement is a compound statement if it is made up of two or more smaller statements. The smaller statements are called component statements of the compound statement. - The role of "And", "Or", "There exists" and "For every" in compound statements. - The meaning of implications "If ", "only if ", " if and only if ". A sentence with if p, then q can be written in the following ways. - p implies q (denoted by p ⇒ q) - p is a sufficient condition for q - q is a necessary condition for p - p only if q - ∼q implies ∼p - The contrapositive of a statement p ⇒ q is the statement ∼ q ⇒ ∼p. The converse of a statement p ⇒ q is the statement q ⇒ p. p ⇒ q together with its converse, gives p if and only if q. 3. The following methods are used to check the validity of statements: (i) direct method (ii) contrapositive method (iii) method of contradiction (iv) using a counter example.
