Basic Discrete Mathematics

Logic, Set Theory, and Probability

Richard Kohar

• Verlag: WSPC
• Jahr: 2016
• ISBN:
  9789814730419
• Thema: Mathématiques
• Thema: Probabilités et statistiques

This lively introductory text exposes the student in the humanities to the world of discrete mathematics. A problem-solving based approach grounded in the ideas of George Pólya are at the heart of this book. Students learn to handle and solve new problems on their own. A straightforward, clear writing style and well-crafted examples with diagrams invite the students to develop into precise and critical thinkers. Particular attention has been given to the material that some students find challenging, such as proofs. This book illustrates how to spot invalid arguments, to enumerate possibilities, and to construct probabilities. It also presents case studies to students about the possible detrimental effects of ignoring these basic principles. The book is invaluable for a discrete and finite mathematics course at the freshman undergraduate level or for self-study since there are full solutions to the exercises in an appendix.

Contents:

• Introduction to Logic
• Proofs and Arguments
• Sets and Set Operations
• Infinity
• Elements of Combinatorics
• Sequences and Series
• The Binomial Theorem
• Introduction to Probability
• Random Variables
• Probability Distributions

Readership: Undergraduates in probability and statistics, logic and set theory.
Key Features:

• One of the first books written on discrete mathematics for the humanities audience
• Litte mathematical knowledge is assumed. Great for humanities, liberal arts, and motivated high-school students
• Uses Pólya's approach to problem solving: understand the problem, devise a plan, carry out the plan, and look back
• Full solutions to all exercises included
• Clear layout of text with essential figures

1

Introduction to Logic

And I can confidently assure you that, as far as you may ever have occasion to exercise your reasoning powers upon any subject, a real acquaintance with the art of Logic will abundantly compensate the labor of acquiring it. Nor have I ever met a person unacquainted with it, who could state and maintain his arguments with facility, clearness, and precision.

—Walker (1847, p. 4)

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.

Nothing is better than eternal happiness. A passing grade is better than nothing.

Therefore, a passing grade is better than eternal happiness.

Do you think just a passing grade will give you eternal happiness? Or am I trying to pass off bad logic to you? How would you show this is an incorrect argument?

Thus, the study of logic is the “study of the methods and principles used in distinguishing correct from incorrect arguments” (Copi, 1954). Of course, this definition does not say that you can only make distinctions of correct and incorrect arguments if you have undertaken the subject of logic, for it is not a necessary condition, but in the very least it will help you to distinguish between correct and incorrect arguments.

1.1 Historical Development of Logic

The historical origins of logic date back to antiquity. Many ancient civilizations such as the Greeks, the Persians, the Chinese, and the Indians had conceived the notion of logic in one form or another. Logic was first established as a formal discipline in the West by the Greek philosopher Aristotle (384 BC–322 BC). His method of analyzing and performing logic was dominant until the advent of modern predicate logic in the early 19^th century. The Organon, the collective name used to describe the six texts of Aristotle’s logical work, contains the heart of Aristotle’s treatment of judgment and formal inference. Here, he introduces the basics and terminology of logic such as the proposition and the syllogism.

His system of logic remained highly influential and unchanged until the 19^th century, when English mathematicians George Boole and John Venn (1834–1923) started to derive a system of manipulating symbols instead of manipulating words. The German mathematician Gottfried Wilhelm von Leibniz (1646–1716) had earlier attempted to create a distinctive logical calculus, but the majority of his work on logic remained unpublished until the turn of the 20^th century.

George Boole (1815–1864) was born a shoemaker’s son in Lincoln, England. He was not afforded an education beyond elementary school due to his family’s small income, and hence was entirely self-taught in the areas of Greek and Latin. By age 16, with the necessity of supporting his poverty-stricken parents, Boole took up teaching in elementary schools, and this had led him to open his own school four years later.

Fig. 1.1 George Boole (1815–1864)

The need to master the subject and prepare his students led Boole to delve into the works of the great mathematicians: Newton, Lagrange, and Laplace. At this time, he began to sub...