Importance of Discrete Mathematics in Computer Science

Importance of Discrete Mathematics in Computer Science Computer science is the study of problems, problem solving and the solutions that come out of the problem solving process, B. Miller and D. Ranum (2013). A computer scientist goal is to develop an algorithm, a step by step list of instructions in solving a problem. Algorithms are finite processes that if followed will solve the problem 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. This essay will discuss the importance of discrete mathematics in computer science. Furthermore, it will attempt to provide an understanding of important related mathematical concepts and demonstrate with evidence based research why these concepts are essential in computer science. The essay will be divided into sections. Section one will define and discuss the importance of discrete mathematics. The second section will focus on and discuss discrete structures and relationships with objects. The set theory would be used as an example and will give a brief understanding of the concept. The third section will highlight the importance of mathematical reasoning. Finally, the essay will conclude with an overview of why discrete mathematics is essential in computer science. Discrete Mathematics 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). 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. Discrete Structures 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). Sets 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. 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 â‚ ¬ 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). Mathematical Reasoning 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). 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 (www.cs.odu.edu). Logic is represented by symbols to represent arbitrary statements. For example the following statements are propositions â€Å"grass is green† and â€Å"2 + 2 = 5†. The first proposition has a truth value of â€Å"true† and the second â€Å"false†. According to S. Waner and S. R Constenoble (1996) a proposition is any declarative sentence which is either true or false. 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 â€Å"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 â€Å"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 deve lop 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). Conclusion 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. 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 w ould 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. References A. Purkiss 2014, Lecture 1: Course Introduction and Numerical Representation, Birkbeck University. B. Miller and D. Ranum 2013. Problem Solving with Algorithms and Data Structures: accessed on [18.01.15] Berztiss, A. (1987). A mathematically focused curriculum for computer science. Communications of the ACM, 30 (5), 356–365. Copi, I. M. (1979). Symbolic Logic (5th ed.). New York: Macmillan Core-Plus Mathematics Project 2014: Discrete Mathematics available at http://www.wmich.edu/cpmp/parentresource/discrete.html [accessed on 25.01.14] 6. D W Kucker Notes on Mathematical Logic; University of Maryland, College Park. Available at http://www.math.umd.edu/~dkueker/712.pdf Accessed on [24.01.15] Denning, P. J. (chair). (1989). Computing as a discipline. Communications of the ACM, 32 (1), 9–23. Dijkstra, E. W. (1989). On the cruelty of really teaching computing science. Communications of the ACM, 32 (12), 1398–1404. Discrete and Continuous Data, (2008). Environmental Systems Research Institute, Inc. Available at http://webhelp.esri.com/arcgisdesktop/9.2/index.cfm?TopicName=Discrete%20and%20continuous%20data [accessed on 18.01.15]. Discrete Structures (2010) available at http://www.cs.odu.edu/~toida/nerzic/content/schedule/schedule.html#day3 [accessed on 25.01.15] Edward R. Scheinerman (2000), Mathematics, A Discrete Introduction (Brooks/Cole, Pacific Grove, CA, 2000): xvii–xviii. Cited in W. J. Rapaport (2013). Discrete Structures. What is Discrete Maths? available from http://www.cse.buffalo.edu/~rapaport/191/whatisdiscmath.html-20130629 accessed on [25.01.2015] Galton, A. (1992). Logic as a Formal Method. The Computer Journal 35 (5), 431–440 Gibbs, N. E., Tucker, A. B. (1986). A model curriculum for a liberal arts degree in computer science. Communications of the ACM 29 (3), 202–210 Goldson, D., Reeves, S., Bornat, R. (1993). A review of several programs for the teaching of logic. The Computer Journal, 36 (4), 373–386. Gries, D. (1990). Calculation and discrimination: A more effective curriculum. Communications of the ACM. 34 (3). 44–55. 16. http://www.cs.odu.edu/~toida/nerzic/content/intro2discrete/intro2discrete.html : Introduction to Discrete Structures What’s and Whys IEEE Model Program Committee. (1983). The 1983 IEEE Computer Society Model Program in Computer Science and Engineering. IEEE Computer Society. Educational Activities Board J. Barwise and J. Etchemendy, Language, Proof and Logic, Seven Bridges Press, New York, 2000, ISBN 1-889119-08-3. J. L. Bell Oppositions and Paradoxes in Mathematics and Philosophy available at http://publish.uwo.ca/~jbell/Oppositions%20and%20Paradoxes%20in%20Mathematics2.pdf accessed on [25.01.2015] 20. K. H Rosen 2012 Discrete Mathematics and its Applications, 7edn, Monmouth University. Myers, Jr. J. P. (1990). The Central role of mathematical logic in computer science. SIGCSE Bulletin, 22 (1), 22–26. Ralston, A. (Ed.) (1989). Discrete Mathematics in the First Two Years. MAA Notes No. 15. The Mathematical Association of America. Ralston, A., Shaw, M. (1980). Curriculum 78 Is computer science really that unmathematical? Communications of the ACM, 23 (2), 67–70. Rota, G.-C. (1997). The phenomenology of mathematical proof. Syntheses, 111:183-196. S. Waner S. R. Costenoble (1996) Introduction to Logic. Saiedian, H. (1992). Mathematics of computing. Computer Science Education, 3 (3), 203-221. Shaw, M. (Ed.) (1985). The Carnegie-Mellon Curriculum for Undergraduate Computer Science. New York: Springer-Verlag Sperschneider, V., Antoniou, G. (1991). Logic: A foundation for computer science International Computer Science Series. Reading, MA: Addison- Wesley The National Council of Teachers of Mathematics (2000). Principles and Standards for School Mathematics. Tucker, A. B. (Ed.) (1990). Computing Curricula 1991: Report of the ACM/IEEE-CS Joint Curriculum Task Force Final Draft, December 17. ACM Order Number 201910. IEEE Computer Society Press Order Number 2220

Rite of Encounter :: encounter

Rite of Encounter Rite of Encounter is, initially a very dry and imposing story. The reader is given same information repeatedly, as if it were not received the first time. This redundancy is an insult to the reader. For instance, in the very first line of the story the narrator tells the reader that, "In the third week of his fasting, Singing- Owl found the white man" (258). This information is given quite clearly, yet later the narrator repeats himself by saying, "A dog meant white men" (259). It is not necessary for the narrator to remind the reader. This "spoon-feeding" is insulting to the reader. The narration was also rather dry. There is little description. The story is conveyed to the reader without any details, and quite plainly, the story is simply reported. The omniscient third person narration is also, at times, confusing. The narration occasionally dips from third person to first without any explanation. For example, when Singing- Owl is suffering of dehydration, fatigue, and hunger the n arrator is reporting the condition of the character. Suddenly, the next line reads, "Water. Must get water" (258). It is unclear who says this. Not suprisingly, Bates, employs this strange tactic again to demonstrate Singing- Owl's exhaustion. The narrator comments on Singing- Owl's declining condition, then says, "Perhaps I'm tired. All right. I am tired" (261). Again, the reader is left unassured of who is speaking. This intentional alteration of narration only robs the story of unity. There is, however, one manipulation of the characters which is interesting. Smallpox is characterized beautifully. Giving life to a disease gives life to a story, which, from the beginning, is dragging on without such animation. Smallpox mocks our "hero", Singing- Owl. This tormenting by a naturally inanimate character saturates the story with fantasy and mysticism. The conclusion of the story, unfortunately, leaves the reader with the same sense of disappointment with which it was started. Singing- Owl, rather than becoming a hero, becomes a marionette for Smallpox to control. Singing- Owl breaks down and agrees to bring Smallpox back to the tribe. Even though Singing- Owl does not completely understand the methods of Smallpox, he does understand the negative repercussions. Yet, Singing- Owl grants Smallpox's wish. This event is disappointing to the reader and degrades the main character. Singing- Owl gains some redemption by trying to infect his enemies, but is not effective and is going to die a dishonored man.

Encounters with crocodiles Essay

I found the crocodiles so intresting that i soon got over my loathing for them and yet with possible exception of as they run the hyenas they must be the meanest creatures in the AFRICAN scene.They will never openly attack,and they will not defend their own eggs ; directly you attempt to land ,into the water they go.It was unwise however to stand to close to the river bank especially at dusk the crocodile lunge out at you with his paws ;instead,having observed you carefully from midstream ,hen swim submeged into the shallows ,and with a sudden like moment of his tail he sweeps you into the water.One of the African boys in the park had been taken like this a few days before we arrived ,and it was no rare thing ,we were told ,for baby elephants to meet their end in just this way when they came down to the river to drink. then the mother elephants goes ranging along the bank quite powerless to retaliate.The crocodiles itself shas mortal enemies ,and not many of the sixty or seventy eggs which female lays,like turtle ,in a hole in a sand bank are destined to survive .Having laid her eggs (they are rather like large white goose -eggs),the mother covers up the hole and then sometimes departs. This is the moment for the moniter lizard to creep out of the undergrowth to scrape the sand away and then to gorge himself .Even if the nest remains undiscovered the young crcodiles need a gold deal of luck to survive. They come struggling to the surface of the sand little ten_inch_long rubbery thing ,and make directly for the water , hissing and snapping as they run .On the bank the marabaou stork with the speed of sword_play flicks them into his long bill ; and if you care to watch you can see the wriggling passage of the young crocodile down the bird scraggy throat.Sometimes the mother crocodile will try to defend her young at this perilous moment ,and this is a facinating thing to see.The marabou, with elaborate unconcern stand in about six inches of water waiting for the next tit bit to come swimming by,and from about twenty yards away the mother crocodile watches :just two murderous eyes above the surface of the stream .Then silently she submerges and come up again about ten y ards from marabou .The bird takes no notice .And now ,having again goes down .This time she is coming in for the kill .it is a matter of about two seconds the marabou abstractely and casually takes a backward step . At the same instant the tremendous jaws of the crocodiles come rearing out of the river and snap together in the empty air at the prcise spot where he was standing .green water streaming off her back ,the crocodiles subsides into river again;and the bird steps back to res ume its meal QUESTIONS/ANSWERS: Q1:what enabled the writer to get over his loathing for the crcodiles? Q2:what do crcodiles do when a person attempts to land on the river bank ? Q3:what method does the crocodiles use to capture ots prey?

Who Invented the Green Garbage Bag Harry Wasylyk

The familiar green plastic garbage bag (made from polyethylene) was invented by Harry Wasylyk in 1950. Canadian Inventors Harry Wasylyk Larry Hansen Harry Wasylyk was a Canadian inventor from Winnipeg, Manitoba, who together with Larry Hansen of Lindsay, Ontario, invented the disposable green polyethylene garbage bag. Garbage bags were first intended for commercial use rather than home use, and the new garbage bags were first sold to the Winnipeg General Hospital. Coincidentally, another Canadian inventor, Frank Plomp of Toronto also invented a plastic garbage bag in 1950, however, he was not as successful as Wasylyk and Hansen were. First Home Use - Glad Garbage Bags Larry Hansen worked for the Union Carbide Company in Lindsay, Ontario, and the company bought the invention from Wasylyk and Hansen. Union Carbide manufactured the first green garbage bags under the name Glad Garbage bags for home use in the late 1960s. How Garbage Bags are Made Garbage bags are made from low-density polyethylene, which was invented in 1942. Low-density polyethylene is soft, stretchy, and water and air proof. Polyethylene is delivered in the form of small resin pellets or beads. By a process called extrusion, the hard beads are converted into bags of plastic. The hard polyethylene beads are heated to a temperature of 200 degrees centigrade. The molten polyethylene is put under high pressure and mixed with agents that provide color and make the plastic pliable. The prepared plastic polyethylene is blown into one long tube of bagging, which is then cooled, collapsed, cut to the right individual length, and sealed on one end to make a garbage bag. Biodegradable Garbage Bags Since their invention, plastic garbage bags have been filling our landfills and unfortunately, most plastics take up to one thousand years to decompose. In 1971, University of Toronto chemist Doctor James Guillet invented a plastic that decomposed in a reasonable time when left in direct sunlight. James Guillet patented his invention, which turned out to be the millionth Canadian patent to be issued.