The Importance of discrete mathematics in computer science


 Charla Lane
 1 years ago
 Views:
Transcription
1 The Importance of discrete mathematics in computer science Exemplified by deductive logic and Boolean algebra By Mathias Schilling BSc Computing, Birkbeck, University of London
2 Preface Discrete mathematics (DM) is a branch of mathematics which deals with mathematical operations within a subset of the real numbers where data is considered more as objects rather than numbers. Characteristically DM provides us problem solving solutions for only distinct and countable quantities which stands clearly in contrast to e.g. the mathematical analysis which encompasses continuous functions applied on uncountable and infinitive quantities. The term discrete (Latin discretum) highlights exactly this difference. Even though some sections of DM like e.g. number or graph theory are relatively old, discrete mathematics has been overshadowed for centuries by the continuous mathematics due to the invention of infinitesimal calculus and its multifaceted applications within natural sciences (particularly in physics). It was only during the 20 century that discrete mathematics has become again more and more important. The new computer based possibilities of data processing have opened a new chapter and pushed the development into the field of discrete mathematics as well as into the computer sciences (CS) itself. But what is that exactly gives discrete mathematics such an importance within the CS? To get a better understanding on how discrete maths concepts are related to computer sciences and why it is relevant for computer engineers to have a solid knowledge in this area we will take a look into mathematical logic as one of major DM topics. Page 1 / 9
3 Mathematical logic in computer sciences History Mathematical logic (ML) emphasises the principles of valid reasoning, inference, validity and soundness and, therefore finds applications in mostly all areas of computing. The study of mathematical proof is specifically important within logic, and has applications to automated theorem proofing and formal verification of software. Before diving deeper into the actual context of ML, we will have a brief look into the historical background of ML, in general and deductive reasoning in detail. Logic is a science of the necessary laws of thought, without which no employment of the understanding and the reason takes place. Immanuel Kant, 1785 The origins of logic can be traced back to the works of the great Greek philosopher Aristotle (384 B.C B.C.) who lived during the golden age of Greek culture. He was one of the first denoted thinkers who believed in the concept of logic with reasoning and arguments as fundamentals for knowledge and covered this thesis in his monumental masterpiece Organon 1. One of his main aims was to establish rules that would enable Greek citizens to distinguish arguments that would formally be valid and correct from those which would be invalid and, therefore wrong. He developed a collection of logical rules for what we currently now as deductive reasoning (syllogismos). Two fundamental laws of logic were established by Aristotle. The Law of NonContradiction and the Law of Excluded Middle. The Law of Noncontradiction 2 establishes that 1 (Epp, 2010) 2 (Gottlieb, 2007) Page 2 / 9
4 no proposition can be true and false at the same time. The Law of the Excluded Middle 3 defines that a proposition must be either true or false. Aristotle s logic is well designed in a way that it is possible to determine the validity of an argument regardless of the matter that is being argued. The engine behind his logic is the syllogism with two premises and one conclusion such as for example: All man are mortal. (Premises 1) Socrates is a man. (Premises 2)  Therefore, Socrates is mortal. (Conclusion) In syllogisms the conclusion is deduced from the premises. If we do not know the conclusion Socrates is mortal than the fact that All man are mortal and Socrates is a man can be used as a proof that indeed Socrates is mortal. This is what is known as deductive argument, which means that the conclusion is a necessary result of the premises. The very beauty of a wellexecuted syllogism is that it cannot guide from true premises to a false conclusion. For instance, in the following argument, formalize it by expressing it as All trees are plants. (True) All willows are trees. (True)  Therefore, all willows are trees. (True) All P s are R s. (True) All Q s are P s. (True)  Therefore, all Q s are R s. (True) 3 (Horn, 2006) Page 3 / 9
5 and notice that it is formally valid and that its validity is completely independent of its statements. Let us have a look onto another, more discussable example to illustrate this validity independency. All trees are rectangular. (False) All cats are trees. (False)  Therefore, all cats are rectangular. (False) Of course the argument is highly questionable, but it is formally valid even its premises and conclusions are false. In the same manner a statement which premises are false and conclusion is true can also be formally valid like for example All trees are flowers. (False) All roses are trees. (False)  Therefore, all roses are flowers. (True) Gottfried Leibniz (1646 A.C A.C.), a german philosopher and mathematician, who was intent upon logic, interpreted Aristotle s thesis by using symbols to mechanize deductive reasoning processes in the same way like reasoning process about numbers and their relationships, was mechanized by algebraic notation 4. Boolean algebra In 1847, over hundred years later, Leibniz idea was finally realized by an english, selftaught mathematicians named George Boole ( ). Boole extended the concept from Leibniz. He created the first algebra of logic and published his thoughts on this topic in his works The Mathematical Analysis of Logic (1847) and An Investigation of the Laws of Thought (1854). He found out that the symbols of logic (today known as the subject of symbolic logic) behave exactly like those in algebra and he used algebraic symbols to express 4 (Epp, 2010) Page 4 / 9
6 logical relations. He claimed that only three operations (AND, OR and NOT) are needed to perform all other logical functions. The terminology Boolean algebra was later suggested by Sheffer in Today George Boole (along with Charles Babbage, who designed the first mechanical calculator, confer Difference Engine) is considered as one of the grandfathers of computing. Applying Boolean algebra But George Boole s thoughts seemed to have been sleeping in a drawer until Claude Shannon ( ), an american mathematician and electric engineer, applied them in While Shannon was studying for his master degree at the Institute of Technology in Massachusetts, he discovered an analogy between the operations of switching devices, as in switching circuits, and the operations of logical connectives. Shannon used Boolean algebra in his thesis (A Symbolic Analysis of Relay and Switching Circuits) to design logic circuits using electromechanical relays. Comparable ideas for application came from the russian logician and theoretician Victor Shestakov in 1935, but unfortunately he did not publish his thoughts until 1941 (Shestakov, V.I. Algebra of Two Poles Schemata). Because Shannon s discovery led us more to our actual question on how Boolean algebra as part of discrete mathematics is applied in modern computer we will have a little deeper look into it. 5 (Huntington, 1933) Page 5 / 9
7 (Figure 1) 6 Imagine figure 1 (a) which shows a simple switch. It is obvious, that current can only flow between the two points if the switch is closed. When the switch is open, current cannot flow. Imagine also that this switch would be part of a circuit like in figure 1 (b). The light bulb in the drawing glows if, and only if, current can flow through the circuit and that only happens if, and only if, the switch is closed. (Figure 2) 7 Figure 2 (a) shows an electric circuit with two switches P and Q in series. The light bulb in this example will only turn on, if both switches are closed. The circuit in figure 2 is just slightly different with the switches in parallel alignment. It is obvious that the light bulb in this example turns only on if at least one of the switches is closed. The behaviour of the two figures is summarised like in the following tables (table 1: switches in series; table 2: switches in parallel): Switches Light bulb Switches Light bulb P Q State P Q State Closed Open Off Closed Open On Closed Closed On Closed Closed Off Open Open Off Open Open On Open Closed Off Open Closed Off 6 Drawing taken from (Epp, 2010) 7 Drawing taken from (Epp, 2010) Page 6 / 9
8 Now if we have a closer look onto the first table, we can determine, that if we are replacing the words Closed with True and Open with False the table becomes the truth table for the logical connective AND. If we are doing the same for the second table we are getting the truth table for OR. From this observation we can conclude that figure 2 (a) corresponds to the logical expression P Q and figure 2 (b) corresponds to P Q and Shannon, was indeed one of the first scientists who successfully applied Boolean algebra. Modern age In modern age, computers are based on the binary system of ones and zeros and are still using George Boole s logic concepts from the middle of the nineteenth century. In order to make a computer carry out arithmetically or logical operations, it uses computer circuits. These are built of logical gates and can be made from either electrical or mechanical switches. Both types work in a quite similar way, with the only difference that a mechanical switch uses lever or wheel positions and electrical switches voltage and current to represent binary digits. A bit sequence can express numbers and letters. A logical gate takes one or more inputs from a circuit and computes based on the underlying Boolean algebra exact one output value. It can also store a result into a so called flipflop (storage unit made from several gates). There are only three different basic types of logical gates AND, OR and NOT. All others such as NAND, NOR, EOR or for example ENOR gates can be made from the basic ones. Page 7 / 9
9 As time has gone by, symbolic logic and Boolean algebra have become the theoretical fundament for many domains in computer science such as artificial intelligence, automata theory, computability, digital logic circuit design and relational database theory. His work was theoretical; he was not an engineer and never actually built a computer or calculating machine. However, Boole s symbolic logic provided the perfect mathematical model for switching theory and for the design of digital circuits. 8 While reviewing the above mentioned works, it has been identified what an essential role deductive logic and Boolean algebra play in the history of computer science. At present time, it is frequent to find logical expressions based on Boolean algebra and deductive logic in form of conditions in the control structures of algorithms and computer programs as well as commands exempli gratia in querying databases. I would like to close this essay by stating that the more we look into a topic, the more questions arise from it and the more chances appear to challenge us in a way that will make us work further in order to improve technology. In line with that, what I have observed while researching, is that most of the time, challenges bring us to discoveries which we would have never even imagined that were possible if we hadn t been put under that pressure or extreme situation that brought us to question the status quo where we stood. I think that, ultimately, technology improvements are the result of challenging great minds, as history has already proven us. 8 (O'Regan, 2012) Page 8 / 9
10 Bibliography Epp, S. (2010). Discrete Mathematics with Applications (4 ed.). Boston: Wadsworth Publishing Co Inc. Gottlieb, P. (2007). Aristotle on Noncontradiction. Stanford Encyclopedia of Philosophy. Retrieved from Horn, L. R. (2006). Contradiction. Stanford Encyclopedia of Philosophy. Retrieved from Huntington, E. V. (1933). New sets of independent postulates for the algebra of logic, with special reference to Whitehead and Russell's Principia mathematica. Retrieved from O'Regan, G. (2012). A Brief History of Computing. Springer. Page 9 / 9
The History of Logic. Aristotle (384 322 BC) invented logic.
The History of Logic Aristotle (384 322 BC) invented logic. Predecessors: Fred Flintstone, geometry, sophists, presocratic philosophers, Socrates & Plato. Syllogistic logic, laws of noncontradiction
More informationLogic in Computer Science: Logic Gates
Logic in Computer Science: Logic Gates Lila Kari The University of Western Ontario Logic in Computer Science: Logic Gates CS2209, Applied Logic for Computer Science 1 / 49 Logic and bit operations Computers
More informationCHAPTER 2. Logic. 1. Logic Definitions. Notation: Variables are used to represent propositions. The most common variables used are p, q, and r.
CHAPTER 2 Logic 1. Logic Definitions 1.1. Propositions. Definition 1.1.1. A proposition is a declarative sentence that is either true (denoted either T or 1) or false (denoted either F or 0). Notation:
More informationTheory of Automated Reasoning An Introduction. AnttiJuhani Kaijanaho
Theory of Automated Reasoning An Introduction AnttiJuhani Kaijanaho Intended as compulsory reading for the Spring 2004 course on Automated Reasononing at Department of Mathematical Information Technology,
More informationTHE FUNDAMENTAL THEOREM OF ARITHMETIC
Introduction In mathematics, there are three theorems that are significant enough to be called fundamental. The first theorem, of which this essay expounds, concerns arithmetic, or more properly number
More information(Refer Slide Time: 05:02)
Discrete Mathematical Structures Dr. Kamala Krithivasan Department of Computer Science and Engineering Indian Institute of Technology, Madras Lecture  1 Propositional Logic This course is about discrete
More informationStudents in their first advanced mathematics classes are often surprised
CHAPTER 8 Proofs Involving Sets Students in their first advanced mathematics classes are often surprised by the extensive role that sets play and by the fact that most of the proofs they encounter are
More informationCS 441 Discrete Mathematics for CS Lecture 2. Propositional logic. CS 441 Discrete mathematics for CS. Course administration
CS 441 Discrete Mathematics for CS Lecture 2 Propositional logic Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Course administration Homework 1 First homework assignment is out today will be posted
More informationBQS 1 Introduction to Artificial Intelligence
Wu Dian September 8, 2010 Professor Craig Graci Csc416 BQS 1 Introduction to Artificial Intelligence This assignment is answering a series of questions after reading chapter 1 of Ben Coppin s Artificial
More informationPredicate logic Proofs Artificial intelligence. Predicate logic. SET07106 Mathematics for Software Engineering
Predicate logic SET07106 Mathematics for Software Engineering School of Computing Edinburgh Napier University Module Leader: Uta Priss 2010 Copyright Edinburgh Napier University Predicate logic Slide 1/24
More informationDigital Logic Circuits
Digital Logic Circuits Digital describes any system based on discontinuous data or events. Typically digital is computer data or electronic sampling of an analog signal. Computers are digital machines
More informationCircuits and Boolean Expressions
Circuits and Boolean Expressions Provided by TryEngineering  Lesson Focus Boolean logic is essential to understanding computer architecture. It is also useful in program construction and Artificial Intelligence.
More informationIntroduction. Logic. Most Difficult Reading Topics. Basic Logic Gates Truth Tables Logical Functions. COMP370 Introduction to Computer Architecture
Introduction LOGIC GATES COMP370 Introduction to Computer Architecture Basic Logic Gates Truth Tables Logical Functions Truth Tables Logical Expression Graphical l Form Most Difficult Reading Topics Logic
More informationModule 3 Digital Gates and Combinational Logic
Introduction to Digital Electronics, Module 3: Digital Gates and Combinational Logic 1 Module 3 Digital Gates and Combinational Logic INTRODUCTION: The principles behind digital electronics were developed
More informationCS61c: Representations of Combinational Logic Circuits
CS61c: Representations of Combinational Logic Circuits J. Wawrzynek October 12, 2007 1 Introduction In the previous lecture we looked at the internal details of registers. We found that every register,
More informationComputation Beyond Turing Machines
Computation Beyond Turing Machines Peter Wegner, Brown University Dina Goldin, U. of Connecticut 1. Turing s legacy Alan Turing was a brilliant mathematician who showed that computers could not completely
More informationDiscrete Mathematics Lecture 1 Logic of Compound Statements. Harper Langston New York University
Discrete Mathematics Lecture 1 Logic of Compound Statements Harper Langston New York University Administration Class Web Site http://cs.nyu.edu/courses/summer05/g22.2340001/ Mailing List Subscribe at
More informationHandout #1: Mathematical Reasoning
Math 101 Rumbos Spring 2010 1 Handout #1: Mathematical Reasoning 1 Propositional Logic A proposition is a mathematical statement that it is either true or false; that is, a statement whose certainty or
More informationUnit 2: Number Systems, Codes and Logic Functions
Unit 2: Number Systems, Codes and Logic Functions Introduction A digital computer manipulates discrete elements of data and that these elements are represented in the binary forms. Operands used for calculations
More informationFundamentals of Mathematics Lecture 6: Propositional Logic
Fundamentals of Mathematics Lecture 6: Propositional Logic GuanShieng Huang National Chi Nan University, Taiwan Spring, 2008 1 / 39 Connectives Propositional Connectives I 1 Negation: (not A) A A T F
More informationChapter I Logic and Proofs
MATH 1130 1 Discrete Structures Chapter I Logic and Proofs Propositions A proposition is a statement that is either true (T) or false (F), but or both. s Propositions: 1. I am a man.. I am taller than
More informationThe Philosophical Importance of Mathematical Logic Bertrand Russell
The Philosophical Importance of Mathematical Logic Bertrand Russell IN SPEAKING OF "Mathematical logic", I use this word in a very broad sense. By it I understand the works of Cantor on transfinite numbers
More informationIt is not immediately obvious that this should even give an integer. Since 1 < 1 5
Math 163  Introductory Seminar Lehigh University Spring 8 Notes on Fibonacci numbers, binomial coefficients and mathematical induction These are mostly notes from a previous class and thus include some
More informationDiscrete mathematics is the study of techniques, ideas and modes
CHAPTER 1 Discrete Systems Discrete mathematics is the study of techniques, ideas and modes of reasoning that are indispensable in applied disciplines such as computer science or information technology.
More informationPRINCIPLE OF MATHEMATICAL INDUCTION
Chapter 4 PRINCIPLE OF MATHEMATICAL INDUCTION Analysis and natural philosophy owe their most important discoveries to this fruitful means, which is called induction Newton was indebted to it for his theorem
More informationCS 3719 (Theory of Computation and Algorithms) Lecture 4
CS 3719 (Theory of Computation and Algorithms) Lecture 4 Antonina Kolokolova January 18, 2012 1 Undecidable languages 1.1 ChurchTuring thesis Let s recap how it all started. In 1990, Hilbert stated a
More informationA Few Basics of Probability
A Few Basics of Probability Philosophy 57 Spring, 2004 1 Introduction This handout distinguishes between inductive and deductive logic, and then introduces probability, a concept essential to the study
More informationClassical Information and Bits
p. 1/24 Classical Information and Bits The simplest variable that can carry information is the bit, which can take only two values, 0 or 1. Any ensemble, description, or set of discrete values can be quantified
More informationSYMBOL AND MEANING IN MATHEMATICS
,,. SYMBOL AND MEANING IN MATHEMATICS ALICE M. DEAN Mathematics and Computer Science Department Skidmore College May 26,1995 There is perhaps no other field of study that uses symbols as plentifully and
More informationDISCRETE MATHEMATICS W W L CHEN
DISCRETE MATHEMATICS W W L CHEN c W W L Chen, 1982, 2008. This chapter originates from material used by the author at Imperial College, University of London, between 1981 and 1990. It is available free
More informationComputer Science. 19. Combinational Circuits. Computer Science. Building blocks Boolean algebra Digital circuits Adder circuit. Arithmetic/logic unit
PA R T I I : A L G O R I T H M S, M A C H I N E S, a n d T H E O R Y PA R T I I : A L G O R I T H M S, M A C H I N E S, a n d T H E O R Y Computer Science 9. Combinational Circuits Computer Science 9.
More information6. Combinational Circuits. Building Blocks. Digital Circuits. Wires. Q. What is a digital system? A. Digital: signals are 0 or 1.
Digital Circuits 6 Combinational Circuits Q What is a digital system? A Digital: signals are or analog: signals vary continuously Q Why digital systems? A Accurate, reliable, fast, cheap Basic abstractions
More informationIncompleteness and Artificial Intelligence
Incompleteness and Artificial Intelligence Shane Legg Dalle Molle Institute for Artificial Intelligence Galleria 2, MannoLugano 6928, Switzerland shane@idsia.ch 1 Introduction The implications of Gödel
More information1. True or False? A voltage level in the range 0 to 2 volts is interpreted as a binary 1.
File: chap04, Chapter 04 1. True or False? A voltage level in the range 0 to 2 volts is interpreted as a binary 1. 2. True or False? A gate is a device that accepts a single input signal and produces one
More informationLecture 3. Mathematical Induction
Lecture 3 Mathematical Induction Induction is a fundamental reasoning process in which general conclusion is based on particular cases It contrasts with deduction, the reasoning process in which conclusion
More informationLecture Notes in Discrete Mathematics. Marcel B. Finan Arkansas Tech University c All Rights Reserved
Lecture Notes in Discrete Mathematics Marcel B. Finan Arkansas Tech University c All Rights Reserved 2 Preface This book is designed for a one semester course in discrete mathematics for sophomore or junior
More informationCS324Artificial Intelligence
CS324Artificial Intelligence Lecture 1: Introduction Waheed Noor Computer Science and Information Technology, University of Balochistan, Quetta, Pakistan Waheed Noor (CS&IT, UoB, Quetta) CS324Artificial
More informationMATH20302 Propositional Logic. Mike Prest School of Mathematics Alan Turing Building Room
MATH20302 Propositional Logic Mike Prest School of Mathematics Alan Turing Building Room 1.120 mprest@manchester.ac.uk April 10, 2015 Contents I Propositional Logic 3 1 Propositional languages 4 1.1 Propositional
More information2011, The McGrawHill Companies, Inc. Chapter 4
Chapter 4 4.1 The Binary Concept Binary refers to the idea that many things can be thought of as existing in only one of two states. The binary states are 1 and 0 The 1 and 0 can represent: ON or OFF Open
More informationWhy Are Computers So Dumb? Philosophy of AI. Are They Right? What Makes us Human? Strong and Weak AI. Weak AI and the Turing Test
Why Are Computers So Dumb? Philosophy of AI Will Machines Ever be Intelligent? AI has made some pretty small strides! A huge number crunching computer beat Kasparov at chess (once) but it still wouldn
More informationQuantification 1 The problem of multiple generality
Quantification 1 The problem of multiple generality Rob Trueman rt295@cam.ac.uk Fitzwilliam College, Cambridge 4/11/11 What is a quantifier? All, some, most, there is one, there are two, there are three...
More informationCHAPTER 10: PROGRAMMABLE LOGIC CONTROLLERS
CHAPTER 10: PROGRAMMABLE LOGIC CONTROLLERS 10.1 Introduction The National Electrical Manufacturers Association (NEMA) as defines a programmable logic controller: A digitally operating electronic apparatus
More informationDirect Proofs. CS 19: Discrete Mathematics. Direct Proof: Example. Indirect Proof: Example. Proofs by Contradiction and by Mathematical Induction
Direct Proofs CS 19: Discrete Mathematics Amit Chakrabarti Proofs by Contradiction and by Mathematical Induction At this point, we have seen a few examples of mathematical proofs. These have the following
More informationRegular Languages and Finite Automata
Regular Languages and Finite Automata 1 Introduction Hing Leung Department of Computer Science New Mexico State University Sep 16, 2010 In 1943, McCulloch and Pitts [4] published a pioneering work on a
More informationInference Rules and Proof Methods
Inference Rules and Proof Methods Winter 2010 Introduction Rules of Inference and Formal Proofs Proofs in mathematics are valid arguments that establish the truth of mathematical statements. An argument
More informationCSE140: Midterm 1 Solution and Rubric
CSE140: Midterm 1 Solution and Rubric April 23, 2014 1 Short Answers 1.1 True or (6pts) 1. A maxterm must include all input variables (1pt) True 2. A canonical product of sums is a product of minterms
More informationTutorial C1. Conversion Between Number Bases. Objectives: Reading: Background: Binary Numbers
utorial C Conversion Between Number Bases Objectives: Reading: After this tutorial you should be able to convert decimal numbers to binary numbers (both unsigned and 2 s complement), and vice versa. You
More informationFACULTY OF INFORMATION TECHNOLOGY
FACULTY OF INFORMATION TECHNOLOGY Course Specifications: (MATH 251) Month, Year: Fall 2008 University: Misr University for Science and Technology Faculty : Faculty of Information Technology Course Specifications
More informationIntroduction to Computer Architecture
Why, How, and What for??? Department of Computer Science Indian Institute of Technology New Delhi, India Outline Introduction 1 Introduction 2 3 What is Computer Architecture? Figure 1: Courtesy: www.psychologytoday.com
More informationArtificial Intelligence. Dr. Onn Shehory Site:
Artificial Intelligence Dr. Onn Shehory email: onn@il.ibm.com Site: www.cs.biu.ac.il/~shechory/ai Outline Administrativa Course overview What is AI? Historical background The state of the art Administrativa
More informationRecursion Theory in Set Theory
Contemporary Mathematics Recursion Theory in Set Theory Theodore A. Slaman 1. Introduction Our goal is to convince the reader that recursion theoretic knowledge and experience can be successfully applied
More informationCHAPTER 3. Methods of Proofs. 1. Logical Arguments and Formal Proofs
CHAPTER 3 Methods of Proofs 1. Logical Arguments and Formal Proofs 1.1. Basic Terminology. An axiom is a statement that is given to be true. A rule of inference is a logical rule that is used to deduce
More informationTest Questions for learning objectives. (Select the best answer from the choices given)
Chapter 1 Test Questions for learning objectives. (Select the best answer from the choices given) 11. Write the definition of a PLC 1) PLCs are designed for use in the control of a wide variety of manufacturing
More information4 BOOLEAN ALGEBRA AND LOGIC SIMPLIFICATION
4 BOOLEAN ALGEBRA AND LOGIC SIMPLIFICATION BOOLEAN OPERATIONS AND EXPRESSIONS Variable, complement, and literal are terms used in Boolean algebra. A variable is a symbol used to represent a logical quantity.
More informationChapter 4. Gates and Circuits. Chapter Goals. Chapter Goals. Computers and Electricity. Computers and Electricity. Gates
Chapter Goals Chapter 4 Gates and Circuits Identify the basic gates and describe the behavior of each Describe how gates are implemented using transistors Combine basic gates into circuits Describe the
More informationCHAPTER 3 Number System and Codes
CHAPTER 3 Number System and Codes 3.1 Introduction On hearing the word number, we immediately think of familiar decimal number system with its 10 digits; 0,1, 2,3,4,5,6, 7, 8 and 9. these numbers are called
More informationUndergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics. Introductory Notes in Discrete Mathematics
Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics Introductory Notes in Discrete Mathematics Marcel B. Finan c All Rights Reserved Last Updated April 6, 2016 Preface
More informationSwitches and Transistors
Switches and Transistors CS 350: Computer Organization & Assembler Language Programming A. Why? It s natural to use on/off switches with voltages representing binary data. Transistor circuits act as switches.
More information2.1.1 Examples of Sets and their Elements
Chapter 2 Set Theory 2.1 Sets The most basic object in Mathematics is called a set. As rudimentary as it is, the exact, formal definition of a set is highly complex. For our purposes, we will simply define
More information2110711 THEORY of COMPUTATION
2110711 THEORY of COMPUTATION ATHASIT SURARERKS ELITE Athasit Surarerks ELITE Engineering Laboratory in Theoretical Enumerable System Computer Engineering, Faculty of Engineering Chulalongkorn University
More informationCS 4700: Foundations of Artificial Intelligence
CS 4700: Foundations of Artificial Intelligence Instructor: Prof. Selman selman@cs.cornell.edu Introduction (Reading R&N: Chapter 1) Course Administration (separate slides) ü What is Artificial Intelligence?
More informationFlorida State University Course Notes MAD 2104 Discrete Mathematics I
Florida State University Course Notes MAD 2104 Discrete Mathematics I Florida State University Tallahassee, Florida 323064510 Copyright c 2011 Florida State University Written by Dr. John Bryant and Dr.
More informationCARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE
CARDINALITY, COUNTABLE AND UNCOUNTABLE SETS PART ONE With the notion of bijection at hand, it is easy to formalize the idea that two finite sets have the same number of elements: we just need to verify
More information22C:19 Discrete Math. So. What is it? Why discrete math? Fall 2009 Hantao Zhang
22C:19 Discrete Math Fall 2009 Hantao Zhang So. What is it? Discrete mathematics is the study of mathematical structures that are fundamentally discrete, in the sense of not supporting or requiring the
More information31 is a prime number is a mathematical statement (which happens to be true).
Chapter 1 Mathematical Logic In its most basic form, Mathematics is the practice of assigning truth to welldefined statements. In this course, we will develop the skills to use known true statements to
More informationMATH HISTORY: POSSIBLE TOPICS FOR TERM PAPERS
MATH HISTORY: POSSIBLE TOPICS FOR TERM PAPERS Some possible seeds for historical developmental topics are: The Platonic Solids Solution of nth degree polynomial equation (especially quadratics) Difference
More information1.10 (a) Effects of logic gates AND, OR, NOT on binary signals in a processor
Chapter 1.10 Logic Gates 1.10 (a) Effects of logic gates AND, OR, NOT on binary signals in a processor Microprocessors are the central hardware that runs computers. There are several components that make
More informationTraditional versus modern logic
1 3 Aristotle s procedure in logic (4th c. BC) Traditional versus modern logic Wilfrid Hodges Herons Brook, Sticklepath, Okehampton November 2010 http://wilfridhodges.co.uk Step One: the raw argument Salt
More informationGates, Circuits, and Boolean Algebra
Gates, Circuits, and Boolean Algebra Computers and Electricity A gate is a device that performs a basic operation on electrical signals Gates are combined into circuits to perform more complicated tasks
More informationLOGICAL INFERENCE & PROOFs. Debdeep Mukhopadhyay Dept of CSE, IIT Madras
LOGICAL INFERENCE & PROOFs Debdeep Mukhopadhyay Dept of CSE, IIT Madras Defn A theorem is a mathematical assertion which can be shown to be true. A proof is an argument which establishes the truth of a
More informationMathematical Induction
Mathematical Induction In logic, we often want to prove that every member of an infinite set has some feature. E.g., we would like to show: N 1 : is a number 1 : has the feature Φ ( x)(n 1 x! 1 x) How
More informationPROOF AND PROBABILITY(teaching notes)
PROOF AND PROBABILITY(teaching notes) In arguing for God s existence, it is important to distinguish between proof and probability. Different rules of reasoning apply depending on whether we are testing
More informationBoolean Design of Patterns
123 Boolean Design of Patterns Basic weave structures interlacement patterns can be described in many ways, but they all come down to representing the crossings of warp and weft threads. One or the other
More informationIntroduction Number Systems and Conversion
UNIT 1 Introduction Number Systems and Conversion Objectives 1. Introduction The first part of this unit introduces the material to be studied later. In addition to getting an overview of the material
More information1.1 What is AI? 1.1 What is AI? Foundations of Artificial Intelligence. 1.2 Acting Humanly. 1.3 Thinking Humanly. 1.4 Thinking Rationally
Foundations of Artificial Intelligence February 22, 2016 1. Introduction: What is Artificial Intelligence? Foundations of Artificial Intelligence 1. Introduction: What is Artificial Intelligence? Malte
More informationPrimes in Sequences. Lee 1. By: Jae Young Lee. Project for MA 341 (Number Theory) Boston University Summer Term I 2009 Instructor: Kalin Kostadinov
Lee 1 Primes in Sequences By: Jae Young Lee Project for MA 341 (Number Theory) Boston University Summer Term I 2009 Instructor: Kalin Kostadinov Lee 2 Jae Young Lee MA341 Number Theory PRIMES IN SEQUENCES
More informationThe Calculus of Logic
The Calculus of Logic George Boole [Cambridge and Dublin Mathematical Journal, Vol. III (1848), pp. 183 98] In a work lately published 1 I have exhibited the application of a new and peculiar form of Mathematics
More informationL  Standard Letter Grade P  Pass/No Pass Repeatability: N  Course may not be repeated
Course: MATH 26 Division: 10 Also Listed As: 200930, INACTIVE COURSE Short Title: Full Title: DISCRETE MATHEMATIC Discrete Mathematics Contact Hours/Week Lecture: 4 Lab: 0 Other: 0 Total: 4 4 Number of
More informationELG3331: Lab 3 Digital Logic Circuits
ELG3331: Lab 3 Digital Logic Circuits What does Digital Means? Digital describes any system based on discontinuous data or events. Typically digital is computer data or electronic sampling of an analog
More informationChapter 1. Computation theory
Chapter 1. Computation theory In this chapter we will describe computation logic for the machines. This topic is a wide interdisciplinary field, so that the students can work in an interdisciplinary context.
More informationThe Foundations: Logic and Proofs. Chapter 1, Part III: Proofs
The Foundations: Logic and Proofs Chapter 1, Part III: Proofs Rules of Inference Section 1.6 Section Summary Valid Arguments Inference Rules for Propositional Logic Using Rules of Inference to Build Arguments
More information1/9. Locke 1: Critique of Innate Ideas
1/9 Locke 1: Critique of Innate Ideas This week we are going to begin looking at a new area by turning our attention to the work of John Locke, who is probably the most famous English philosopher of all
More informationAnnex A1 COURSE DESCRIPTIONS AND LEARNING OBJECTIVES FOR:
Annex A1 COURSE DESCRIPTIONS AND LEARNING OBJECTIVES FOR: Discrete Math (Already in ACGM, course description has been revised, and learning outcomes have been added.) Electrical Circuits I (Already in
More informationSCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics. Numbers
SCHOOL OF ENGINEERING & BUILT ENVIRONMENT Mathematics Numbers. Factors 2. Multiples 3. Prime and Composite Numbers 4. Modular Arithmetic 5. Boolean Algebra 6. Modulo 2 Matrix Arithmetic 7. Number Systems
More information2.0 Chapter Overview. 2.1 Boolean Algebra
Thi d t t d ith F M k 4 2 Boolean Algebra hapter Two Logic circuits are the basis for modern digital computer systems. To appreciate how computer systems operate you will need to understand digital logic
More information7.1 An Axiomatic Approach to Mathematics
Chapter 7 The Peano Axioms 7.1 An Axiomatic Approach to Mathematics In our previous chapters, we were very careful when proving our various propositions and theorems to only use results we knew to be true.
More informationAdversary Modelling 1
Adversary Modelling 1 Evaluating the Feasibility of a Symbolic Adversary Model on Smart Transport Ticketing Systems Authors Arthur Sheung Chi Chan, MSc (Royal Holloway, 2014) Keith Mayes, ISG, Royal Holloway
More informationMath Department Student Learning Objectives Updated April, 2014
Math Department Student Learning Objectives Updated April, 2014 Institutional Level Outcomes: Victor Valley College has adopted the following institutional outcomes to define the learning that all students
More informationEastern Washington University Department of Computer Science. Questionnaire for Prospective Masters in Computer Science Students
Eastern Washington University Department of Computer Science Questionnaire for Prospective Masters in Computer Science Students I. Personal Information Name: Last First M.I. Mailing Address: Permanent
More informationSwitching Circuits & Logic Design
Switching Circuits & Logic Design JieHong Roland Jiang 江介宏 Department of Electrical Engineering National Taiwan University Fall 2013 1 Course Info Instructor JieHong R. Jiang office: 242 EEII office
More informationIf f is a 11 correspondence between A and B then it has an inverse, and f 1 isa 11 correspondence between B and A.
Chapter 5 Cardinality of sets 51 11 Correspondences A 11 correspondence between sets A and B is another name for a function f : A B that is 11 and onto If f is a 11 correspondence between A and B,
More informationSo let us begin our quest to find the holy grail of real analysis.
1 Section 5.2 The Complete Ordered Field: Purpose of Section We present an axiomatic description of the real numbers as a complete ordered field. The axioms which describe the arithmetic of the real numbers
More informationIt is time to prove some theorems. There are various strategies for doing
CHAPTER 4 Direct Proof It is time to prove some theorems. There are various strategies for doing this; we now examine the most straightforward approach, a technique called direct proof. As we begin, it
More informationSwitching Algebra and Logic Gates
Chapter 2 Switching Algebra and Logic Gates The word algebra in the title of this chapter should alert you that more mathematics is coming. No doubt, some of you are itching to get on with digital design
More informationLecture 1: Elementary Number Theory
Lecture 1: Elementary Number Theory The integers are the simplest and most fundamental objects in discrete mathematics. All calculations by computers are based on the arithmetical operations with integers
More informationTh e ontological argument distinguishes itself from the cosmological
Aporia vol. 18 no. 1 2008 Charles Hartshorne and the Ontological Argument Joshua Ernst Th e ontological argument distinguishes itself from the cosmological and teleological arguments for God s existence
More informationLIFELONG LEARNING MAKING DISCRETE MATH RELEVANT FOR INFORMATION SYSTEMS PROFESSIONALS
LIFELONG LEARNING MAKING DISCRETE MATH RELEVANT FOR INFORMATION SYSTEMS PROFESSIONALS Dr. David F. Wood, Robert Morris University, wood@rmu.edu Dr. Valerie J. Harvey, Robert Morris University, harvey@rmu.edu
More informationPost s (and Zuse s) forgotten ideas in the Philosophy and History of Computer Science.
Post s and Zuse s forgotten ideas in the Philosophy of Computer Science. L. De Mol Post s (and Zuse s) forgotten ideas in the Philosophy and History of Computer Science. Liesbeth De Mol Boole centre for
More information2. Methods of Proof Types of Proofs. Suppose we wish to prove an implication p q. Here are some strategies we have available to try.
2. METHODS OF PROOF 69 2. Methods of Proof 2.1. Types of Proofs. Suppose we wish to prove an implication p q. Here are some strategies we have available to try. Trivial Proof: If we know q is true then
More informationNOT AND OR XOR NAND NOR
NOT AND OR XOR NAND NOR Expression 1: It is raining today Expression 2: Today is my birthday X Meaning True False It is raining today It is not raining Binary representation of the above: X Meaning 1 It
More information