Basic Discrete Mathematics

The following sections are included:

• Dedication
• Contents
• Preface
• Acknowledgments

The first notion in which a student can form a sense of logic is by viewing it as the examination of reasoning in arguments. Arguments consist of either true or false statements, and from these statements, we can decide if the reasoning that links these statements will yield a true conclusion. By using the knowledge that we know, and drawing conclusions, this process is called logical inference.

The following sections are included:

• What is a Proof?
• Rules of Inference
  • Fundamental Logical Equivalences
  • Logical Equivalences for Implication and Biconditional
• What is an Argument?
• Modus Ponens
• Modus Tollens
• Syllogism
  • Hypothetical Syllogism
  • Disjunctive Syllogism
• Fallacies
  • Affirming the Disjunct
  • Affirming the Consequent
  • Denying the Antecedent
• Deductive Reasoning and Inductive Reasoning
• More Complex Arguments
• Truth and Validity
• Review
• Bibliographic Remarks

The following sections are included:

• Introduction to Sets
• Russell's Paradox
• Set Operations
• Principle of Inclusion and Exclusion
• Set Operations Revisited
• Using Venn Diagrams to Check Categorical Syllogisms
• Review
• Bibliographic Remarks

The following sections are included:

• Introduction
  • Euclid and Prime Numbers
• Galileo and Cantor Counting Numbers Leads to Heresy (or The Integers and the Rationals are Countable)
• Hilbert's Hotel
• The Real Numbers are Not Countable
• Review
• Bibliographic Remarks

Combinatorics is the study of counting various combinations or configurations. The ability to count, without directly listing or enumerating, all the combinations or configurations forms part of the basis of probability (see for example, Sec. 8.7 and Sec. 10.2).

The following sections are included:

• Sequences
• Method of Finite Differences
• Series and Sigma Notation
• Arithmetic Series
• Properties of Series
• Pi Notation
• The Geometric Series
• Investment: Applications of the Geometric Series
  • Compounded Interest
  • Annuities
  • Present Value
• The Infinite Geometric Series (A Trojan and a Tortoise)
• Proofs by Mathematical Induction
  • Inductive Reasoning versus Mathematical Induction
  • The Principle of Mathematical Induction
  • Proofs Using the Principle of Mathematical Induction
  • Abusing Mathematical Induction: More Fallacies
  • Guessing the Formula
• Review
• Bibliographic Remarks

The following sections are included:

• Pascal's Triangle
• The Binomial Theorem
• Review
• Bibliographic Remarks

The following sections are included:

• Introduction
• Experiments, Sample Spaces, Sample Points, and Events
• Probability of an Event in a Uniform Sample Space
• Definition and Axioms of Probability
• Simple Probability Rules
• Risk: Applications of Probability
• Using Counting Techniques to Calculate Probability
  • Permutations
  • Combinations
• Conditional Probability
• Independence
  • Independence versus Mutually Exclusive Events
  • The Sally Clark Case
• The Origins of Probability: Gambling
• Bayes' Theorem
• Review
• Bibliographic Remarks

The following sections are included:

• Introduction
• Random Variables & Probability Distributions
• Expected Value
• Variance & Standard Deviation
• Review

In this chapter, we will look at probability distributions that arise frequently when investigating certain kinds of experiments.

The following sections are included:

• Appendix A: Probability Distribution Tables
• Appendix B: Prerequisite Knowledge
• Appendix C: Solutions to Exercises

The following sections are included:

• Bibliography
• Index