This text aims to give an introduction to select topics in discrete mathematics at a level appropriate for first or second year undergraduate math majors, especially those who intend to teach middle and high school mathematics. The book began as a set of notes for the Discrete Mathematics course at the University of Northern Colorado. This course serves both as a survey of the topics in discrete math and as the "bridge" course for math majors, as UNC does not offer a separate "introduction to proofs" course. Most students who take the course plan to teach, although there are a handful of students who will go on to graduate school or study applied math or computer science. For these students the current text hopefully is still of interest, but the intent is not to provide a solid mathematical foundation for computer science, unlike the majority of textbooks on the subject. Discrete mathematics forms the mathematical foundation of computer and information science. It is also a fascinating subject in itself. Learners will become familiar with a broad range of mathematical objects like sets, functions, relations, graphs, that are omnipresent in computer science. Perhaps more importantly, they will reach a certain level of mathematical maturity – being able to understand formal statements and their proofs; coming up with rigorous proofs themselves; and coming up with interesting results. This book attempts to be rigorous without being overly formal. This means, for every concept we introduce we will show at least one interesting and non-trivial result and give a full proof. However, we will do so without too much formal notation, employing examples and figures whenever possible.
James Franklin is Professor in the School of Mathematics and Statistics, University of New South Wales, Sydney, Australia. He is the author of An Aristotelian Realist Philosophy of Mathematics, Proof in Mathematics: An Introduction, and books on the history of probability, Australian philosophy, and knowledge in science. He holds BA (Hons) in pure mathematics, Sydney University, 1974, MA (Hons), Sydney University, 1976 PhD in Mathematics, Warwick University, 1982. In the philosophy of mathematics, James leads the "Sydney School", which defends an Aristotelian realist theory of mathematics. Starting from applied mathematics, this theory takes mathematics to be primarily about the quantitative and structural properties of the real world, such as ratios, symmetry and continuity. In extreme risks, where probabilities have to be evaluated beyond the range of the available data, James recommends "advocacy methods", by which expert opinion is subject to sceptical checking by a panel of experts with a different agenda. In the philosophy of probability, James's historical book on the pre-mathematical theory of probability informs his defence of an objective Bayesian view of probability, according to which logical probability is an objective logical relation between evidence and conclusion. In ethics, James was awarded the 2005 Eureka Prize for Research in Ethics for work on the parallel between objectivity in mathematics and in ethics. He edited a book on Catholic social justice theory and maintained a website on indigenous violence issues. He conducted the Restraint Project, a research project on the virtue of temperance in the Australian context. His work on the foundations of ethics is The Worth of Persons: The Foundations of Ethics.
| Preface Chapter 1. Foundations of Discrete Mathematics Introduction to Discrete Mathematics Types of Sets: Finite, Infinite, Power Sets Functions and Their Types Cardinality of Sets Recursion and Recurrence Relations Chapter 2. Logic and Proof Techniques Propositional Logic Logical Connectives and Truth Tables Predicate Logic and Quantifiers Rules of Inference Methods of Proof: Direct, Contradiction Chapter 3. Combinatorics Overview of Combinatorics Permutations and Combinations The Pigeonhole Principle Binomial Cofficients and Pascal’s Triangle Inclusion-Exclusion Principle Chapter 4. Number Theory Introduction to Number Theory Divisibility and Prime Numbers Modular Arithmetic Fermat’s Little Theorem and Euler’s Theorem Chapter 5. Relations and Functions Relations: Definition and Properties Properties of Relations (Reflexive, Symmetric, Transitive) Equivalence Relations and Partitions Partial Orders and Lattices Hasse Diagrams Composition of Functions |
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