Finite Mathematics

An Introduction with Applications in Business, Social Sciences, and Music

1st Edition - April 24, 2025
Imprint: Academic Press
Author: Andrew McHugh
Language: English
Paperback ISBN:
9 7 8 - 0 - 4 4 3 - 2 9 0 9 4 - 7
eBook ISBN:
9 7 8 - 0 - 4 4 3 - 2 9 0 9 5 - 4

Finite Mathematics: An Introduction with Applications in Business, Social Sciences, and Music presents core concepts of finite mathematics in a clear, intuitive fashion design… Read more

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Finite Mathematics: An Introduction with Applications in Business, Social Sciences, and Music presents core concepts of finite mathematics in a clear, intuitive fashion designed to reinforce understanding. The book begins with finite mathematics foundations, with explanations and exercises on combinatorics, logic, set theory, sequences and series, functions and functional notation, elementary probability, linear programming and systems, and Markov chains. Later chapters explore and explain a range of finite mathematics applications, from game theory to voting, apportionment, finance, graph theory, and the science and physics of music.

Written with an accessible, example-based approach, this book engages STEM and non-STEM students alike, preparing them for courses across a range of quantitative fields, social sciences, and the liberal arts. Problem-solving exercises are featured at the conclusion of each subsection, with corresponding answers in the appendix. The book is also accompanied by a solutions manual, sample projects assignments, tests, lecture slides, and datasets on a companion website.

Title of Book
Cover image
Title page
Table of Contents
Copyright
Dedication
Biography
Preface
Chapter 1: Logic
1.1. Statements
1.1.1. What is a statement?
1.1.2. Exercises
1.2. Combining statements together with “and” and “or”
1.2.1. Truth tables for “and” and “or”
1.2.2. Exercises
1.3. Negation, ∼
1.3.1. Defining the negation and some notation and examples
1.3.2. Negating statements beginning with “all” or “some”
1.3.3. Exercises
1.4. The order of operations in logic
1.4.1. Exercises
1.5. Truth tables and De Morgan's laws
1.5.1. Making truth tables
1.5.2. Exercises
1.6. Conditional statements
1.6.1. What is a conditional statement?
1.6.2. The truth table for p→q
1.6.3. The converse, the inverse, and the contrapositive
1.6.4. Exercises
Chapter 2: Set theory
2.1. Sets, elements, set notation
2.1.1. The undefinition of a set of elements
2.1.2. Set notation – the roster method
2.1.3. Sets of numbers
2.1.4. Set builder notation
2.1.5. Venn diagrams
2.1.6. Subsets
2.1.7. Exercises
2.2. Intersection, union, and complements of sets
2.2.1. The intersection of sets
2.2.2. The union of two sets
2.2.3. The complement of a set
2.2.4. Interval notation
2.2.5. Exercises
2.3. The cardinality of finite sets
2.3.1. The cardinality of finite sets
2.3.2. The cardinality of a union of finite sets
2.3.3. Exercises
Chapter 3: Combinatorics
3.1. Permutations, factorials, and the fundamental counting principle
3.1.1. The election problem and the fundamental counting principle
3.1.2. Permutations
3.1.3. Factorials
3.1.4. Exercises
3.2. Combinations
3.2.1. Learning about combinations
3.2.2. Exercises
3.3. Using the fundamental counting principle
3.3.1. Exercises
Chapter 4: Sequences and summations
4.1. Sequences
4.1.1. Mathematical sequence notation
4.1.2. Exercises
4.2. Sigma summation notation
4.2.1. What is Sigma notation?
4.2.2. Summation of constants
4.2.3. Basic arithmetic sums
4.2.4. Linearity properties of Sigma summation
4.2.5. Exercises
4.3. Geometric sums
4.3.1. Finite geometric sums
4.3.2. Infinite geometric sums
4.3.3. Exercises
4.4. Pascal's triangle and the binomial theorem
4.4.1. Binomial expansions using Pascal's triangle
4.4.2. Exercises
Chapter 5: Functions
5.1. Functional notation
5.1.1. Hands-on approach to functional notation
5.1.2. Exercises
5.2. Graphing functions
5.2.1. The Cartesian coordinate system
5.2.2. The graph of a function
5.2.3. Using technology to graph functions
5.2.4. Exercises
5.3. The domain, codomain, and image of a function
5.3.1. The domain of a function
5.3.2. The codomain of a function
5.3.3. The domain→codomain notation for a function
5.3.4. The image of a function
5.3.5. Exercises
5.4. Linear functions
5.4.1. Constant functions
5.4.2. Linear functions
5.4.3. Exercises
5.5. Piecewise functions
5.5.1. Exercises
5.6. Functions of two or more variables
5.6.1. Graphing functions of two variables
5.6.2. Functions of three or more variables
5.6.3. Exercises
Chapter 6: Elementary probability theory
6.1. The framework of probability
6.1.1. The experiment
6.1.2. The sample space
6.1.3. Events
6.1.4. Probability of an event on a finite sample space with equally likely outcomes
6.1.5. Some probability rules
6.1.6. Probability on finite sample spaces without equally likely outcomes
6.1.7. Exercises
6.2. Probability on infinite sample spaces
6.2.1. Probability on infinite discrete sample spaces
6.2.2. Probability on continuous sample spaces
6.2.3. Exercises
6.3. Conditional probability and independent events
6.3.1. Conditional probability
6.3.2. Independent events
6.3.3. Exercises
6.4. Random variables
6.4.1. Probability distributions for discrete random variables
6.4.2. Probability distributions for continuous random variables
6.4.3. Cumulative probability distribution functions
6.4.4. Exercises
6.5. Expectation values
6.5.1. Exercises
6.6. The binomial distribution
6.6.1. Bernoulli trials
6.6.2. Binomial experiments and the binomial distribution
6.6.3. Exercises
6.7. The normal distribution
6.7.1. The standard normal distribution
6.7.2. General normal distributions
6.7.3. Exercises
Chapter 7: Elementary statistics
7.1. Populations and data
7.1.1. Gathering data
7.1.2. Types of data and levels of measurements
7.1.3. Exercises
7.2. Presenting data graphically
7.2.1. Presenting qualitative data graphically: bar and pie charts
7.2.2. Graphically presenting quantitative data: histograms and scatter plots
7.2.3. Exercises
7.3. Measures of central tendency: mean, median, and mode
7.3.1. The mean
7.3.2. The median
7.3.3. The mode
7.3.4. More thoughts on the mean, median, and mode
7.3.5. Exercises
7.4. Measures of dispersion, part 1: range and standard deviation
7.4.1. The range of a set of data
7.4.2. The variance and the standard deviation of census data for a population
7.4.3. The variance and standard deviation of a probability distribution
7.4.4. A nice formula for the variance of a census data set or a probability distribution
7.4.5. The variance and standard deviation of a sample data set
7.4.6. Exercises
7.5. Measures of dispersion, part 2: quartiles and percentiles
7.5.1. Quartiles
7.5.2. Percentiles
7.5.3. Exercises
Chapter 8: Linear systems and matrices
8.1. Review of solving and graphing linear equations in one and two variables
8.1.1. Review of solving linear equations in one variable
8.1.2. Linear equations in two variables
8.1.3. Exercises
8.2. Linear equations in three and more variables
8.2.1. Linear equations in three variables
8.2.2. Linear equations in more than three variables
8.2.3. Exercises
8.3. Solving systems of linear equations in two variables
8.3.1. Solving systems of equations by substitution
8.3.2. Solving linear systems in two variables using elimination
8.3.3. Solving linear systems in two variables graphically
8.3.4. Exercises
8.4. Solving systems of linear equations in three and more variables
8.4.1. Solving linear systems of equations in three variables using substitution
8.4.2. Solving systems of linear equations in three variables using elimination
8.4.3. Exercises
8.5. Solving systems of linear equations using row reduction on augmented matrices
8.5.1. Representing systems of linear equations as augmented matrices
8.5.2. Elementary row operations and row reduction of augmented matrices
8.5.3. Exercises
8.6. Matrices
8.6.1. Operations on matrices
8.6.2. Matrix multiplication
8.6.3. The transpose of a matrix
8.6.4. Square matrices and inverse matrices
8.6.5. Determinants and Cramer's rule
8.6.6. Exercises
Chapter 9: Linear programming and the simplex method
9.1. Graphing linear inequalities and systems of linear inequalities
9.1.1. Graphing linear inequalities in two dimensions
9.1.2. Systems of linear inequalities in two dimensions
9.1.3. Graphing feasible regions in two dimensions
9.1.4. Exercises
9.2. Linear programming
9.2.1. Linear programming in two dimensions
9.2.2. Exercises
9.3. The simplex method
9.3.1. A common form of a linear programming problem
9.3.2. Slack variables
9.3.3. The simplex method on a linear programming problem in common form
9.3.4. Exercises
Chapter 10: Markov chains
10.1. Introduction to Markov chains
10.1.1. A visit to the zoo
10.1.2. An example of shifting demographics
10.1.3. Exercises
10.2. Absorbing states and limiting behavior
10.2.1. A simple version of the gambler's ruin problem
10.2.2. Absorbing states and absorbing Markov chains
10.2.3. Calculating the limiting behavior of an absorbing Markov chain
10.2.4. Exercises
10.3. Regular Markov chains
10.3.1. Definition of a regular Markov chain
10.3.2. Limiting behavior of regular Markov chains
10.3.3. Exercises
Chapter 11: Game theory
11.1. Introduction to game theory
11.1.1. What is game theory?
11.1.2. Two-person games and zero-sum games
11.1.3. Saddle point solutions and strictly determined games
11.1.4. Exercises
11.2. Mixed strategies and optimal strategies
11.2.1. Mixed strategies
11.2.2. Optimal mixed strategies for zero-sum games with a 2×2 payoff matrix and no saddle point solution
11.2.3. The expected payoff as a vector-matrix-vector product
11.2.4. Exercises
Chapter 12: Mathematics of voting
12.1. Voter profiles and methods of voting
12.1.1. Voter profiles
12.1.2. Methods of voting: the plurality method
12.1.3. The Borda method
12.1.4. The pairwise comparison method
12.1.5. Instant run-off method
12.1.6. Exercises
12.2. Voting fairness criteria and Arrow's theorem
12.2.1. Voting fairness criteria: the majority criterion
12.2.2. The Condorcet criterion
12.2.3. The monotonicity criterion
12.2.4. Irrelevant alternatives criterion
12.2.5. Arrow's theorem
12.2.6. Exercises
Chapter 13: Mathematics of apportionment
13.1. Methods of apportionment
13.1.1. The challenge of apportionment
13.1.2. The Hamilton method
13.1.3. The Jefferson method
13.1.4. The Webster method
13.1.5. The Huntington-Hill method
13.1.6. Exercises
13.2. Criteria for apportionment
13.2.1. The quota rule
13.2.2. The Alabama paradox
13.2.3. The population paradox
13.2.4. The new states paradox
13.2.5. Exercises
Chapter 14: Mathematics of finance
14.1. Simple and compounded interest
14.1.1. Simple interest
14.1.2. Compounded interest
14.1.3. Continuous compounding
14.1.4. Exercises
14.2. Amortization
14.2.1. Amortization: mortgages, car loans, and payment plans
14.2.2. Exercises
Chapter 15: Introduction to graph theory
15.1. Graphs
15.1.1. Graphs - vertices and edges
15.1.2. The degree of a vertex and the degree sum formula
15.1.3. The complete graph on n vertices, Kn
15.1.4. Exercises
15.2. Walks, trails, paths, circuits, and cycles
15.2.1. Walks
15.2.2. Trails and circuits
15.2.3. Paths and cycles
15.2.4. Euler trails and Euler circuits
15.2.5. Hamilton paths and Hamilton cycles
15.2.6. Exercises
15.3. Coloring a graph
15.3.1. Graph theory and making maps
15.3.2. Coloring of a graph
15.3.3. The Welsh-Powell algorithm for coloring a graph
15.3.4. Scheduling and coloring graphs
15.3.5. Exercises
Chapter 16: Mathematics and music
16.1. Sound waves and sine waves
16.1.1. Sound waves
16.1.2. The mathematics of sine waves
16.1.3. Graphing sine waves using computer technology
16.1.4. Exercises
16.2. Sine waves and the elements of music
16.2.1. Frequency and pitch
16.2.2. Amplitude and dynamics
16.2.3. Piecewise functions and rhythm
16.2.4. Wave shape and timbre
16.2.5. Exercises
16.3. Musical scales and chords
16.3.1. Major scales
16.3.2. Minor scales
16.3.3. Chromatic scales
16.3.4. Chords
16.3.5. Exercises
Appendix A: Answers to selected exercises
A.1. Chapter 1
A.2. Chapter 2
A.3. Chapter 3
A.4. Chapter 4
A.5. Chapter 5
A.6. Chapter 6
A.7. Chapter 7
A.8. Chapter 8
A.9. Chapter 9
A.10. Chapter 10
A.11. Chapter 11
A.12. Chapter 12
A.13. Chapter 13
A.14. Chapter 14
A.15. Chapter 15
A.16. Chapter 16
Index

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