Shen Hua , Zhang Sheng Yuan - "university education" --2013
Is a mathematics course on discrete mathematics in nature, is an important part of the structure and quality of students math mathematical knowledge. Mathematical discipline system, while very large, but can be divided into continuous, discrete and stochastic three categories. In most science and engineering curriculum design, mathematics courses typically include: Advanced Mathematics, linear algebra, discrete mathematics, probability theory and mathematical statistics and so on. Advanced mathematics to provide continuous processing mathematical tools of mathematical problems needed; Linear Algebra and Discrete Mathematics provides mathematical tools to deal with the problem of discrete mathematics; and probability and statistics provides mathematical tools to deal with stochastic mathematical problems.
As Mrs Pan pointed out in the article: As a computer -disciplinary tool, discrete modeling discrete mathematics difference simply place higher mathematics, but also discrete place closely associated with the Mathematics and Computer, but also the discrete mathematics as one of the reasons computer science core curriculum. From the student perspective, discrete mathematics is abstract, conceptual and more fragmented knowledge, etc., in a study likely to encounter difficulties, which greatly affected the enthusiasm of their learning. This article explores the nature of mathematics discrete mathematics, the purpose is to straighten out the relationship between these concepts and knowledge, and thus achieve the purpose of solving students' learning difficulties.
The contents include discrete mathematics and mathematical logic, set theory, graph theory and algebraic structure of four parts, which set theory part plays pivot role. Mathematical logic and set theory two parts content if handled well, teaching the entire curriculum will play a crucial role. Has been part of the research papers on teaching mathematical logic and set theory to discuss, this in-depth analysis of mathematical logic and set theory of teaching content, clarify the nature of mathematics and their mutual links , clarify teaching ideas. Teaching practice shows that these analyzes enable teaching in the teaching process of teaching content main line clear, clear teaching objectives, so as to effectively improve the quality of mathematics teaching quality and students.
First, the mathematical nature of mathematical logic part
one, part of the mathematical nature of propositional logic is mathematical logic.
In the teaching process, before the introduction of the teaching of propositional logic, can allow students to more "human" and "computer" respective strengths. Most students can come to this conclusion: people longer than "smart" and the computer longer than "computing." So, let the computer growth "intelligence", the main direction is to "smart" computerized: by the "smart" thinking problem into a problem determination by calculation. The intelligence is based on logical reasoning, so "smart" computerized mathematical logic is to first of all. Therefore, mathematical logic is an important basis for the computer's "Artificial Intelligence."
In essence propositional logic discrete mathematics logic is part of this mathematical, logical or specifically algebra. Algebraic methods are the basic elements of the object and the operation, the basic process algebraic mode: symbolized (objects), operations, legal operation, calculation, standard application. This way of thinking just to remind students to review Algebra content in high school learned can be quickly accepted. Look propositional logic part of this teaching content, according to a mode that is substantially expanded: propositional symbol of the (target), logical operations (connective) calculates law (substantially equivalent formula), equivalent calculation, standard (Paradigm ), application (solution of the problem is determined to prove equivalence formula, the actual application, reasoning, etc.). Thus, knowledge of propositional logic is not part of this scattered throughout the algebra of the main line.
The practice shows this part of the series of propositional logic by logic algebra main content main line, clear teaching objectives, can be a good teaching effect; while students can learn to understand algebraic way of thinking, to improve their quality of math and applied Mathematics ability to solve practical problems.
In the process of teaching propositional logic, in addition to emphasis on algebraic way of thinking it must also be emphasized that the "standard model" (paradigm) is the core of this section. On the one hand the ultimate goal of the paradigm is the equivalent of calculus, the other paradigm is between a bridge between propositional formula and truth table, and therefore has a very high theoretical and practical value.
Second, the mathematical nature of predicate logic part is to introduce the idea of variables and functions.
From a mathematical essence, predicate logic is the idea of introducing a logical variables and functions. In this vision, those basic concepts becomes clear: the individual variables are variables, the predicate is a function of the individual domain is defined fields, properties predicates are functions of one variable, the relationship between the predicate is a multi-function .... Then again the process algebraically: symbolized (verb), operation (connective) calculates law (mainly an increase of quantifiers equivalent formula), equivalent calculation, standard (toe paradigm), the application (problem determination, proof equivalent type, the practical application, the predicate logic theory).
Of course, predicate logic content than the deep and complex propositional logic, discrete mathematics undergraduate, this can only be regarded as part of the basis of predicate logic.
Second, the nature of mathematical set theory part of the
usual set theory, discrete mathematics section also contains two chapters: the basic concepts of set theory, binary relations and functions. Since the secondary stage has simple content set theory, so this part of the students will not feel strange.
Mathematical set theory is the cornerstone of the whole, almost all mathematical concepts can be expressed in the language of set theory, mathematical form an independent scientific system on the basis of set theory. In fact, basically build process can also be seen from the mathematics of this scientific system and a set of binary relations this part.
This chapter first set theory is a process algebra: objects (sets), operation (set operation), the arithmetic law (set identity), calculus, applications (counts prove identity, practical application). Here the lack of a standard type, calculus is actually a collection of standard type can have, but here there is no logical calculus of the standard paradigm just as important. It can be seen from the content and structure, set theory and propositional logic part of the two are very similar, which will be explored in later.
With this basic set of language, you can define the binary relation. Followed by calculation of the operational nature of the relationship (which is part of the algebraic method). Then the three special relationship: equivalence relation, and partial order function. Meaning equivalence relation is "classified," one of the basic idea of this method is both mathematics is a common task of data mining; and the significance of partial order is to "sort", which is the most basic computer algorithm study.
With the definition of the function, analytics can be mounted thereon; After the binary operation function definition, then substituting with the basic mathematics. With analytics, algebra, mathematics, science and the basic framework of the system will basically set up better.
This is the set theory of mathematics. From the collection to the relationship, to the function and operation, built-disciplinary basic mathematics. This is the essence of this part of the mathematical set theory. These clear, the teacher can do chest with the "number", the overall situation. The students are introduced to these mathematical nature, students can initially understand the structure, meaning and value of this part, this part of the learning and mastering is of great help. And after learning two parts, students become familiar with and grasp the ideas and methods of algebra, abstract algebra part of the follow-up study on psychological knowledge and have a certain amount of preparation.
Third, the inherent mathematical logic and set theory, the basic content of the links
mentioned before, propositional logic and set theory has two parts content very similar. Specifically, the arithmetic operation of the two parts has a strong correspondence relationship law. For example, logical operations {~, ∧, ∨} { ~, ∩, ∪} -one correspondence between the set operation. Most teachers can recognize this and use it in teaching. For example, remind students to observe the law of correspondence between arithmetic operations and set operations of legal logic operation of law in teaching arithmetic logic operation, which helps students understand and master the law of arithmetic logic operations. Before some discrete mathematics textbook also on this part of the set theory and mathematical logic, although doing so undermines the logic - Collection - Algebra continuity of the case, but the students have a preliminary knowledge of set theory began, and then use the set teaching theory and propositional logic similar to the logic of aid on the part of the content, but also has its merits at.
In fact, with propositional logic and tool set operation can be deduced and Binary Operations: a given set of A and B, is assumed Collection E. For any given element x∈E, denoted by p proposition "x∈A", q represents the proposition x∈B, the propositional propositional formula ~ p represents the "x∈ ~ A", p∧q indicates "x∈ B∩A ", p∨q represents" x∈A∪B ". This is a logical operation corresponds to the conversion relationship in the set operation. Further, (tautology) 1 indicates x∈E never really style, never fake formula (contradiction) 0 means "x∈Φ", then set theory can be directly derived from the basic formula of propositional logic equivalence Some basic identities (such as associative, commutative, distributive property, de Morgan law, etc.).
Of course, you want to explore more deeply the relationship between the equivalent type of set theory identities and logical operations, the need to use predicate logic tools, but this is beyond the scope of teaching and research, so this is not further elaborate expansion, interested readers can explore on their own.
IV Conclusion
I believe that emphasizing the application of discrete mathematics in computer science is necessary to help improve student learning enthusiasm and application awareness, but the nature of mathematics discrete mathematics itself should not be overlooked carried, after all, it is an important mathematical courses. This article deals with the teaching of mathematical logic and set theory module of some mathematical nature of discrete mathematics, aimed at teaching discrete mathematics fully integrate their application on the basis of its grasp on the nature of mathematics, which can effectively improve teaching effectiveness, and We can develop students' algebraic thinking habits, improve their discrete modeling capabilities.
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Author: hjlweilong
Source: CNBLOGS
Original: https://www.cnblogs.com/hjlweilong/p/9493751.html
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