本文是计算机专业的留学生作业写作范例，题目是“Importance of Discrete Mathematics in Computer Science（离散数学在计算机科学中的重要性）”，离散数学研究的是具有离散值的结构，这些结构通常在本质上是无限的。正如实数系统在连续数学中起着至关重要的作用一样，整数也是离散数学的基石。离散数学为分析在不同状态下变化的现实世界现象提供了优秀的建模工具，是广泛应用的重要工具，从计算机到电话路由，从人员分配到遗传学。
Abstract 摘要
Discrete mathematics is concerned with structures which take on a discrete value often infinite in nature. Just as the real-number system plays a crucial role in continuous mathematics, integers are the cornerstone in discrete mathematics. Discrete mathematics provides excellent modelling tools for analysing real-world phenomena that varies in one state or another and is a vital tool used in a wide range of applications, from computers to telephone call routing and from personnel assignments to genetics, E.R. Scheinerman (2000) cited in W. J. Rapaport 2013).
The difference between discrete mathematics and other disciplines is the basic foundation on proof as its modus operandi for determining truth, whereas science for example, relies on carefully analysed experience. According to J. Barwise and J. Etchemendy, (2000), a proof is any reasoned argument accepted as such by other mathematicians. Discrete mathematics is the background behind many computer operations (A. Purkiss 2014, slide 2) and is therefore essential in computer science. According to the National Council of Teachers of Mathematics (2000), discrete mathematics is an essential part of the educational curriculum (Principles and Standards for School Mathematics, p. 31). K. H Rosen (2012) cites several important reasons for studying discrete mathematics including the ability to comprehend mathematical arguments. In addition he argues discrete mathematics is the gateway to advanced courses in mathematical sciences.
离散数学和其他学科的区别在于，证明是确定真理的基本方法，而科学则依赖于仔细分析的经验。根据J. Barwise和J. Etchemendy的观点，证明是被其他数学家接受的任何理性论证。离散数学是许多计算机操作的背景，因此在计算机科学中是必不可少的。根据全国数学教师委员会(2000)，离散数学是教育课程的重要组成部分(《学校数学原则和标准》，第31页)。K. H Rosen(2012)列举了学习离散数学的几个重要原因，包括理解数学论点的能力。此外，他认为离散数学是通往数学科学高级课程的大门。
1.Discrete Mathematics离散数学
According to K. H. Rosen, (2012) discrete mathematics has more than one purpose but more importantly it equips computer science students with logical and mathematical skills. Discrete mathematics is the study of mathematics that underpins computer science, with a focus on discrete structures, for example, graphs, trees and networks, K H Rosen (2012). It is a contemporary field of mathematics widely used in business and industry. Often referred to as the mathematics of computers, or the mathematics used to optimize finite systems (Core-Plus Mathematics Project 2014). It is an important part of the high school mathematics curriculum. Discreet mathematics is a branch of mathematics dealing with objects that can assume only distinct separated values (mathworld wolfram.com). Discrete mathematics is used in contrast with continuous mathematics, a branch of mathematics dealing with objects that can vary smoothly including calculus (mathworld wolfram.com). Discrete mathematics includes graph theory, theory of computation, congruences and recurrence relations to name but a few of its associated topics (mathworld wolfram.com).
根据K. H. Rosen，(2012)离散数学有不止一个目的，但更重要的是它为计算机科学学生提供了逻辑和数学技能。离散数学是计算机科学基础的数学研究，关注离散结构，例如图，树和网络，K H Rosen(2012)。它是一个现代数学领域，广泛应用于商业和工业。通常被称为计算机的数学，或用于优化有限系统的数学(Core-Plus数学项目2014)。它是高中数学课程的重要组成部分。离散数学是处理只能假设不同值的对象的数学分支。离散数学是与连续数学相对的，连续数学是数学的一个分支，处理对象可以平滑变化，包括微积分(mathworldwolfram.com)。离散数学包括图论、计算理论、同余和递归关系等一些相关主题(mathworldwolfram.com)。
Discrete mathematics deals with discrete objects which are separated from each other. Examples of discrete objects include integers, and rational numbers. A discrete object has known and definable boundaries which allows the beginning and the end to be easily identified. Other examples of discrete objects include buildings, lakes, cars and people. For many objects, their boundaries can be represented and modelled as either continuous or discrete, (Discrete and Continuous Data, 2008). A major reason discrete mathematics is essential for the computer scientist, is, it allows handling of infinity or large quantity and indefiniteness and the results from formal approaches are reusable.
2.Discrete Structures离散结构
To understand discrete mathematics a student must have a firm understanding of how to work with discrete structures. These discrete structures are abstract mathematical structures used to represent discrete objects and relationships between these objects. The discrete objects include sets, relations, permutations and graphs. Many important discrete structures are built using sets which are collections of objects K H Rosen (2012).
要理解离散数学，学生必须牢固地理解如何处理离散结构。这些离散的结构是抽象的数学结构，用于表示离散的对象和这些对象之间的关系。离散对象包括集合、关系、排列和图。许多重要的离散结构是使用集合来构建的，这些集合是对象的集合K H Rosen(2012)。
3.Sets集
As stated by Cantor (1895: 282) cited in J. L. Bell (1998) a set is a collection of definite, well- differentiated objects. K. H Rosen (2012) states discrete structures are built using sets, which are collections of objects used extensively in counting problems; relations, sets of ordered pairs that represent relationships between objects, graphs, sets of vertices and edges that connect vertices and edges that connect vertices; and finite state machines, used to model computing machines. Sets are used to group objects together and often have similar properties.
正如J. L. Bell(1998)引用的Cantor(1895: 282)所说，集合是一组明确的、有很好区别的对象的集合。K. H Rosen(2012)指出离散结构是用集合构建的，集合是在计算问题中广泛使用的对象的集合;关系，表示对象、图形、连接顶点和连接顶点的边的顶点和边之间关系的有序对的集合;有限状态机，用于模拟计算机。集合用于将对象分组在一起，通常具有相似的属性。
For example, all employees working for the same organisation make up a set. Furthermore those employees who work in the accounts department form a set that can be obtained by taking the elements common to the first two collections. A set is an unordered collection of objects, called elements or members of the set. A set is said to contain its elements. To denote that a is an element of the set A, we write a € A. For example the set O of odd positive integers less than 10 can be expressed by O = {1, 3, 5, 7, 9}. Another example is, {x |1 ≤ x ≤ 2 and x is a real number.} represents the set of real numbers between 1 and 2 and {x | x is the square of an integer and x ≤ 100} represents the set {0. 1, 4, 9, 16, 25, 36, 49, 64, 81, 100}, (www.cs.odu.edu).
4.Mathematical Reasoning数学推理
Logic is the science for reasoning, Copi, (1971) and a collection of rules used in carrying out logical reasoning. The foundation for logic was laid down by the British mathematician George Boole. Logic is the basis of all mathematical reasoning and of all automated reasoning. It has practical applications to the design of computing machines, to the specification of systems, to artificial intelligence, to computer programming, to programming languages and to other areas of computer science, K H Rosen, (2012 page 1).
逻辑是推理的科学，Copi(1971)和一组用于进行逻辑推理的规则。逻辑的基础是由英国数学家乔治·布尔奠定的。逻辑是所有数学推理和所有自动推理的基础。它在计算机设计、系统规范、人工智能、计算机编程、编程语言和计算机科学的其他领域都有实际应用，K H Rosen(2012年第1页)。
Mathematical logic, starts with developing an abstract model of the process of reasoning in mathematics, D. W. Kucker page 1. Following the development of an abstract model a study of the model to determine some of its properties is necessary. The aim of logic in computer science is to develop languages to model the situations we encounter as computer science professionals, in such a way that we can reason about them formally. Reasoning about situations means constructing arguments about them; we want to do this formally, so that the arguments are valid and can be defended rigorously, or executed on a machine.
In understanding mathematics we must understand what makes a correct mathematical argument, that is, a proof. As stated by C. Rota (1997) a proof is a sequence of steps which leads to the desired conclusion Proofs are used to verify that computer programs produce the correct result, to establish the security of a system and to create artificial intelligence.
Logic is interested in true or false statements and how the truth or falsehood of a statement can be determined from other statements . Logic is represented by symbols to represent arbitrary statements. For example the following statements are propositions “grass is green” and “2 + 2 = 5”. The first proposition has a truth value of “true” and the second “false”. According to S. Waner and S. R Constenoble (1996) a proposition is any declarative sentence which is either true or false.
逻辑感兴趣的是真实或虚假的陈述，以及如何从其他陈述中判断一个陈述的真实或虚假。逻辑是用符号来表示任意语句的。例如，下面的陈述是命题“草是绿的”和“2 + 2 = 5”。第一个命题的真值是“真”，第二个是“假”。根据S. Waner和S. R . Constenoble(1996)的观点，命题是任何陈述句，要么是真，要么是假。
Many in the computing community have expressed the view that logic is an essential topic in the field of computer science (e.g., Galton, 1992; Gibbs & Tucker, 1986; Sperschneider & Antoniou, 1991). There has also been concern that the introduction of logic to computer science students has been and is being neglected (e.g., Dijkstra, 1989; Gries, 1990). In their article “A review of several programs for the teaching of logic”, Goldson, Reeves and Bornat (1993) stated: There has been an explosion of interest in the use of logic in computer science in recent years. This is in part due to theoretical developments within academic computer science and in part due to the recent popularity of Formal Methods amongst software engineers. There is now a widespread and growing recognition that formal techniques are central to the subject and that a good grasp of them is essential for a practising computer scientist. (p. 373). In his paper “The central role of mathematical logic in computer science”, Myers (1990) provided an extensive list of topics that demonstrate the importance of logic to many core areas in computer science and despite the fact that many of the topics in Myers list are more advanced than would be covered in a typical undergraduate program, the full list of topics covers much of the breadth and depth of the curriculum guidelines for computer science, Tucker (1990). The model program report (IEEE, 1983) described discrete mathematics as a subject area of mathematics that is crucial to computer science and engineering. The discrete mathematics course was to be a pre or co requisite of all 13 core subject areas except Fundamentals of Computing which had no pre requisites. However in Shaw’s (1985) opinion the IEEE program was strong mathematically but disappointing due to a heavy bias toward hardware and its failure to expose basic connections between hardware and software. In more recent years a task force had been set up to develop computer science curricula with the creation of a document known as the Denning Report, (Denning, 1989). The report became instrumental in developing computer science curriculum. In a discussion of the vital role of mathematics in the computing curriculum, the committee stated, mathematical maturity, as commonly attained through logically rigorous mathematics courses is essential to successful mastery of several fundamental topics in computing, (Tucker, 1990, p.27).
It is generally agreed that students in undergraduate computer science programs should have a strong basis in mathematics and attempts to recommend which mathematics courses should be required, the number of mathematics courses and when the courses should be taken have been the source of much controversy (Berztiss, 1987; Dijkstra, 1989; Gries, 1990; Ralston and Shaw, 1980; Saiedian 1992). A central theme in the controversy within the computer science community has been the course discrete mathematics. In 1989, the Mathematical Association of America published a report about discrete mathematics at the undergraduate level (Ralston, 1989). The report made some recommendations including offering discrete mathematics courses with greater emphasis on problem solving and symbolic reasoning (Ralston, 1989; Myers, 1990).
5.Conclusion结论
The paper discussed the importance of discrete mathematics in computer science and its significance as a skill for the aspiring computer scientist. In addition some examples of this were highlighted including its usefulness in modelling tools to analyse real world events. This includes its wide range of applications such as computers, telephones, and other scientific phenomena. The next section looked at discrete structures as a concept of abstract mathematical structures and the development of set theory a sub topic within discrete mathematics.
本文讨论了离散数学在计算机科学中的重要性及其作为一种技能对有抱负的计算机科学家的意义。此外，本文还着重介绍了这方面的一些例子，包括它在建模工具分析现实世界事件时的有用性。这包括其广泛的应用，如计算机、电话和其他科学现象。下一节将离散结构视为抽象数学结构的概念，集合理论的发展是离散数学中的一个子主题
The essay concluded with a literature review of evidence based research in mathematical reasoning where various views and opinions of researchers, academics and other stakeholders were discussed and explored. The review makes clear of the overwhelming significance and evidence stacked in favour for students of computer science courses embarking on discrete mathematics.
Overall, it is generally clear that pursuit of a computer science course would most definitely need the associated attributes in logical thinking skills, problem solving skills and a thorough understanding of the concepts. In addition the review included views of an increased interest in the use of logic in computer science in recent years. Furthermore formal techniques have been acknowledged and attributed as central to the subject of discrete mathematics in recent years.
总的来说，很明显，追求一门计算机科学的课程肯定需要逻辑思维能力、解决问题的能力和对概念的透彻理解。此外，该综述还包括近年来对计算机科学中逻辑的使用越来越感兴趣的观点。此外，形式技术已被承认，并归因于中心的主题离散数学在近年来。
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