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Discrete Mathematics with Coding book cover

1st Edition

Discrete Mathematics with Coding

By Hugo D Junghenn Copyright 2024
    547 Pages 145 Color & 55 B/W Illustrations
    by Chapman & Hall

    547 Pages 145 Color & 55 B/W Illustrations
    by Chapman & Hall
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    This book, for a first undergraduate course in Discrete Mathematics, systematically exploits the relationship between discrete mathematics and computer programming. Unlike most discrete mathematics texts focusing on one of the other, the book explores the rich and important connection between these two disciplines and shows how each discipline reinforces and enhances the other.

    The mathematics in the book is self-contained, requiring only a good background in precalculus and some mathematical maturity. New mathematical topics are introduced as needed.

    The coding language used is VBA Excel. The language is easy to learn, has intuitive commands, and the reader can develop interesting programs from the outset. Additionally, the spreadsheet platform in Excel makes for convenient and transparent data input and output and provides a powerful venue for complex data manipulation. Manipulating data is greatly simplified using spreadsheet features and visualizing the data can make programming and debugging easier.

    The VBA language is seamlessly integrated into the spreadsheet environment with no other resources required. Furthermore, as some of the modules in the book show, intricate patterns, graphs, and animation in the form of moving cells is possible.

    Features

    • Introduces coding in VBA Excel assuming no previous coding experience.

    • Develops programs in Linear Analysis, Logic, Combinatorics, Probability, and Number Theory.

    • Contains over 90 fully tested and debugged programs. The code for these is as well as the exercises is available on the author's website.

    • Contains numerous examples that gradually introduce the reader to coding techniques.

    • Includes programs that solve systems of linear equations, linear programming problems, combinatorial problems, Venn diagram problems and programs that produce truth tables from logic statements and logic statements from switching and gate circuits, encrypt and decrypt messages and simulate probability experiments.

    　

    I

    Essentials of VBA

    1

    Introduction

    1.1 Modules, Subs, and Functions

        1. Data Output and Input. OutputExample,InputExample

        2. Cell Characteristics. CellAttributes

        3. Number Formats. NumberFormat

        4. Passing Arguments. SwitchIntegers

        5. User Defined Data Types

        6. Command Buttons

        7. Spin Buttons. MixColors

        8. Exercises

      1. VBA Operators

        1. Arithmetic Operators

        2. Comparison Operators

        3. Logical Operators

        4. Exercises

      2. Conditional Statements

    3.1

    The If Then Else Statement. Grade

    3.2

    The Select Case Statement. CycleEdges

    3.3

    Exercises

    4 Loops

    4.1

    The For Next Statement

    　

    4.2

    The Do While Loop. ReduceFraction, NumberOfTerms

    4.3

    The Do Until Loop

    4.4

    Exit Statements

    4.5

    Finding the Zeros of a Function. FindZero, FunctionVal

    4.6

    Exercises

    5 Arrays

    5.1

    Declaring Arrays

    5.2

    First and Last Indices of an Array

    5.3

    Passing and Returning Arrays

    　

    　

        1. Variants and the Array Function

        2. Sorting Numeric Data. NumericSort

        3. Finding Nearest Numbers. NearestNumbers

        4. Stacks. Push, Pop

        5. Exercises

      1. String Functions

        1. The VBA Concatenation Operator &

          2. Duplicating Strings. DupString

        2. The VBA Extraction Function Mid

          3. Matching Symbols. IsMatch

          Inserting, Replacing a String. RemoveInsertString

          A Variation of Mid. Midd

        3. The VBA ASCII Functions Asc and Chr

          4. Boolean ASCII Functions. IsUpper,IsLower,IsLetter,IsDigit

        4. VBA Data Conversion Functions CStr, CInt, CLng, CDbl

        5. The VBA Function Replace

          6. Removing Spaces From a String. RemoveWhiteSpace

          Removing Duplicate Characters. RemoveDupChar

        6. The VBA Function InStr

        7. The VBA Operator Like

        8. The VBA Functions Join and Split

          9. Deleting Duplicate Strings. DeleteDupStr

        9. Exercises

      2. Grids

        1. Setting Up a Grid. MakeGrid

        2. The Grid As a Torus. TorusPoint

        3. Installing Coordinates in a Grid. InstallCoord

        4. Sieve of Erastothenes. Sieve

        5. A Changing Rectangle. CollapseExpandRectangle, Delay

        6. Table Sum Game. TableSum

        7. Finding the Nearest Cells. NearestCells

        8. A Growing Spiral. Spiral

        9. Billiard Ball. BilliardBall

        10. Exercises

      3. Recursion

        1. The Factorial Function. FactRecursive

        2. Binary Trees. Binarytree

        3. The Tower of Hanoi. HanoiTower

        *8.4 A Complex Arithmetic Calculator. ComplexCalc

        *8.5 A Polynomial Calculator. PolyCalc

        8.6 Exercises

      4. Charts and Graphs

        1. Frequency Charts. LetterFreqency

        2. Drawing Lines. GraphPoly

        3. Intersection of Two Lines. LineIntersect

        4. Projection of a Point onto a Line. Point2Line

        5. The Incenter of a Triangle. Incenter

        6. Function Graphs. GraphFunction

        7. Fitting a Line to Data. Least Squares

        8. Graphing Parametric Curves. GraphCurve

        9. A Fractal Game. ChaosTriangleGame

        10. Overlaying Graphs. PredatorPrey

        11. Bezier Curves. Bezier

        12. Exercises

    1. Random Numbers

        1. The VBA Function Rnd

          2. Random Decimals. RndDec

          Random Integers. RndInt

          Random Permutations. RndPerm

          Random Position. NextRndPos

          Probability Distributions. RndOut

        2. A Nonsense Generator. Nonsense

        3. Area Under a Curve. AreaUnderCurve

        4. A Simple Random Walk. Random Walk

        5. Spreading Disease. SpreadOfDisease

        6. Particle Swarm Optimization. ParticleSwarmOpt

        7. A Randomly Sprouting Plant. SproutWither

        8. Random Walk of a Rotating Pattern. RotateAndMove

        9. Random Tessellations. RandomTess

        10. Maxwell’s Entropy Demon. MaxwellsDemon

        11. Multi-Random Walk. RandomWalks

        12. A Shedding Organism. RandomWalkShed

        13. A Splitting Organism. RandomWalkSplit

        14. Random Mazes. Maze

        15. Exercises

      II Linear Analysis

    2. Linear Equations

        1. Matrix Arrays. MatrixIn, MatrixOut

        2. Systems of Linear Equations

        3. The Gauss Jordan Method

        4. Row Operations with VBA. RowOperations

        5. Row Echelon Form with VBA. RowEchelon

        6. Exercises

    　

    　

    　

    12 Linear Programming

    12.1 Linear Inequalities and Feasible Regions

    12.2 Formulation of the General LP Problem

    The Standard LP Max Problem

    The Standard LP Min Problem

    Nonstandard LP Problems

        1. Graphic Solution for Two Dimensions

        2. The Simplex Method

        3. Standard Max Problem with VBA. Simplex

        4. Duality and the Standard LP Min Problem

        5. Solving Nonstandard LP Problems

        6. Exercises

      1. Matrix Algebra

      13.1

      Matrix Scalar Multiplication. ScalarMult

      13.2

      Matrix Addition and Substraction. AddSubMat

      13.3

      Matrix Multiplication. MultMat

      13.4

      Inverse of a Matrix. MatrixInversion

      13.5

      Matrix Powers. PowerMat

      13.6

      Input-Output Models. IOModel

      13.7

      Polynomial of Best Fit. PolyFitData

      *13.8

      The Barnsley Fern. AffineIterations

      *13.9

      A Matrix Calculator. MatrixCalculator

      13.10

      Exercises

      　

    1. Determinants

        1. Definition and Properties

        2. Determinants Using VBA. DetEchelon, DetRecursive

        3. Cramer’s Rule. CramersRule

        4. Collinear and Coplanar Points Using Determinants

        5. Areas and Volumes Using Determinants

        6. Circumscribing a Triangle. Circumcenter

        7. Exercises

    　

    　

    III

    Logic

    15 Propositional Logic

    　

    15.1 Compound Statements

        1. Equivalent Statements and Laws of Logic

        2. Truth Tables with VBA. Stmt2TruthTable

        3. Statement from a Truth Table. TruthTable2DisjNormal

        4. Valid Arguments

        5. Exercises

      1. Switching Circuits

        1. Introduction

        2. Series and Parallel Circuits

        3. Equivalent Circuits

        4. Circuit Expressions With VBA. SwitchCircuitExpr

        5. Exercises

    1. Gates and Logic Circuits

        1. The Gates NOT, AND, OR

        2. The Gates XOR, NAND, NOR

        3. Logic Circuit Expressions with VBA. LogicCircuitExpr

        4. Half Adders

        5. VBA Simulation of a Half Adder

        6. Full Adders

        7. VBA Simulation of a 4-bit Adder. FullAdder4

        8. Exercises

      1. Combinatorics

      1. Sets

        1. Introduction

        2. The Union of Two Sets. Union

        3. The Intersection of Two Sets. Intersection, Card

        4. The Complement of a Set. Complement

        5. Extensions to Three or More Sets

        6. Calculating Sets with VBA. SetCalculator

        7. Venn Diagram Components with VBA. VennParts

        8. Laws of Set Equality

        9. Laws of Set Inclusion

        10. Cartesian Products, Relations, and Functions

        11. Exercises

    19 Counting

    19.1

    The Addition Principle

    *19.2

    Venn Solutions with VBA. Venn3Solver

    19.3

    The Multiplication Principle

    19.4

    Permutations

    19.5

    Generating Permutations with VBA. Permutations

    19.6

    Combinations

    19.7

    Generating Combinations with VBA. Combinations

    19.8

    Traveling Salesman Problem. TravelSales

    *19.9

    Permutation Algebra. MultPerm, InvPerm, PowerPerm

    *19.10

    Permutation Cycles. PermCycles

    19.11

    Exercises

    　

    　

      1. Probability

      1. Probability

        1. Sample Spaces, Probabilities, and Events

        2. Throwing a Pair of Fair Dice. DiceRelativFreq

        3. Poker Hands. PokerHands

        4. Drawing Balls from an Urn

        5. Simulation of the Binomial Distribution. GaltonsBoard

        6. Conditional Probability

        7. Bayes’ Theorem

        8. Disease Testing. TruePosTrueNeg

        9. Independence

        10. Exercises

      2. Random Variables

        1. Examples of Random Variables

        2. Expected Value of a Random Variable

        3. Properties of Expectation

        4. Variance and Standard Deviation of a Random Variable

        5. The Law of Large Numbers

          6. Mean Time For Two Heads. MeanTimeForFirstHH

          Bernoulli-Laplace model. BernoulliLaplaceMeanTime

        6. Central Limit Theorem

        7. Exercises

      3. Markov Chains

        1. First Order Transition Probabilities

        2. Higher Order Transition Probabilities

        3. The Initial Distribution of a Markov Chain

        4. The Steady State Vector

        5. Position Dependent Random Walk. PositionRandWalk

        6. Polya Urn Model. PolyaUrn

        7. Ehrenfest Diffusion Model. EhrenfestUrns

        8. Stepping Stone Model. SteppingStone

        9. Exercises

      1. Properties of Numbers

      1. Divisibility and Prime Numbers

        1. The Division Algorithm

        2. Number Bases. BaseToBase

    *23.3 Wolfram’s Binary Rules. WolframRules

        1. Greatest Common Divisor. GCD

        2. Prime Divisibility Property

        3. The Fundamental Theorem of Arithmetic

      1. Prime Decomposition with VBA. PrimeDecomposition

      2. Exercises

      1. Congruence

        1. Definition and Basic Properties

        2. Congruence Arithmetic

        3. Fermat’s Little Theorem

        4. Remainder of a Product. ProductRmdr

        5. Remainder of a Power. PowerRmdr

        6. Solving Congruence Equations. CongruenceSolver

    *24.7 Modular Arithmetic. CongruenceTables

      1. 24.8 ExercisesCryptography

        1. Encoding the Alphabet

        2. Caesar’s Shift Cipher

        3. Zigzag Cipher

        4. Spiral Cipher. SpiralCipher

        5. Affine Cipher. AffCipher

        6. Permutation Cipher. PermCipher

        7. Vigen`ere Cipher. VCipher

        8. Four Square Cipher. FourSquare

        9. Matrix Cipher. MatrixCipher

        10. The RSA Algorithm. RSACipher

        11. Exercises

      1. The Enigma Machine

        1. Description

        2. Implementation in VBA. EnigmaCipher

        3. A Closer Look at the Mathematics

      2. Large Numbers

        1. Arithmetic of Large Positive Integers

          2. Formatting. AddLeadingZeros, RemoveLeadingZeros

          Addition of Positive Integers. AddPosInt

          Subtraction of Positive Integers. SubtractPosInt

          Multiplication of Positive Integers. MultPosInt

          Division Algorithm for Positive Integers. DivAlgPos

        2. Arithmetic of Large Signed Integers

          3. Negative Signs. IsNeg, AbsVal, Neg

          Addition and Subtraction of Integers. AddInt, SubtractInt

          Multiplication of Integers. MultInt, PowerInt

          Division Algorithm for Integers. LargeDivAlg

        3. Arithmetic of Large Decimals

    Decimal Points. SplitDec, AddTrailingZeros, RestoreDecPt, RemoveTrailingZeros

    Decimal Addition and Subtraction. AddDec, SubtractDec

    Decimal Multiplication. MultDec

    Decimal Division. DivDec, LargeRound

    Decimal Exponentiation. PowerDec

    *27.4 A Large Scale Calculator. LargeCalc

    27.5 Exercises

    Index

    Biography

    Hugo D. Junghenn is Professor of Mathematics at The George Washington University. He has published numerous journal articles and is the author of several books, including A Course in Real Analysis and Principles of Analysis: Measure, Integration, Functional Analysis, and Applications. His research interests include functional analysis, semigroups, and probability.

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