Discrete Mathematics: Introduction to Mathematical Reasoning

  • AUTHOR: Susanna S. Epp
  • ISBN-13: 9780495826170 
  • Grade(s): 9 | 10 | 11 | 12
  • 648 Pages  Hardcover 
  • 1st Edition
  • ©2011     Published
  • Prices are valid only in the respective region

Overview

About The Product

Susanna Epp's DISCRETE MATHEMATICS: AN INTRODUCTION TO MATHEMATICAL REASONING, provides the same clear introduction to discrete mathematics and mathematical reasoning as her highly acclaimed DISCRETE MATHEMATICS WITH APPLICATIONS, but in a compact form that focuses on core topics and omits certain applications usually taught in other courses. The book is appropriate for use in a discrete mathematics course that emphasizes essential topics or in a mathematics major or minor course that serves as a transition to abstract mathematical thinking. The ideas of discrete mathematics underlie and are essential to the science and technology of the computer age. This book offers a synergistic union of the major themes of discrete mathematics together with the reasoning that underlies mathematical thought. Renowned for her lucid, accessible prose, Epp explains complex, abstract concepts with clarity and precision, helping students develop the ability to think abstractly as they study each topic. In doing so, the book provides students with a strong foundation both for computer science and for other upper-level mathematics courses.

Features

  • Epp addresses difficulties in understanding logic and language with very concrete and easy-to-conceptualize examples, helping students with a variety of backgrounds better comprehend basic mathematical reasoning, and enabling them to construct sound mathematical arguments.
  • A large number of exercises provide ample practice for students, with numerous applied problems covering an impressive array of applications.
  • Worked examples are developed in intuitive problem-solution format. Proof solutions are developed in two steps, with discussion on how one might come to devise the proof oneself followed by a concise version of the proof,enabling students of different levels to work at their own pace with adequate support and additional help for those who need it.
  • Margin notes highlight issues of particular importance and help students avoid common mistakes.
  • "Test Yourself" questions with answers at the end of each section provide immediate feedback to students regarding their understanding of basic concepts.

About the Contributor

AUTHOR
  • Susanna S. Epp

    Susanna S. Epp received her Ph.D. in 1968 from the University of Chicago, taught briefly at Boston University and the University of Illinois at Chicago, and is currently Vincent DePaul Professor of Mathematical Sciences at DePaul University. After initial research in commutative algebra, she became interested in cognitive issues associated with teaching analytical thinking and proof and has published a number of articles and given many talks related to this topic. She has also spoken widely on discrete mathematics and has organized sessions at national meetings on discrete mathematics instruction. In addition to Discrete Mathematics with Applications and Discrete Mathematics: An Introduction to Mathematical Reasoning, she is co-author of Precalculus and Discrete Mathematics, which was developed as part of the University of Chicago School Mathematics Project. Epp co-organized an international symposium on teaching logical reasoning, sponsored by the Institute for Discrete Mathematics and Theoretical Computer Science (DIMACS), and she was an associate editor of Mathematics Magazine from 1991 to 2001. Long active in the Mathematical Association of America (MAA), she is a co-author of the curricular guidelines for undergraduate mathematics programs: CUPM Curriculum Guide 2004.

Table of Contents

1. SPEAKING MATHEMATICALLY.
Variables. The Language of Sets. The Language of Relations and Functions.
2. THE LOGIC OF COMPOUND STATEMENTS.
Logical Form and Logical Equivalence. Conditional Statements. Valid and Invalid Arguments.
3. THE LOGIC OF QUANTIFIED STATEMENTS.
Predicates and Quantified Statements I. Predicates and Quantified Statements II. Statements with Multiple Quantifiers. Arguments with Quantified Statements.
4. ELEMENTARY NUMBER THEORY AND METHODS OF PROOF.
Direct Proof and Counterexample I: Introduction. Direct Proof and Counterexample II: Rational Numbers. Direct Proof and Counterexample III: Divisibility. Direct Proof and Counterexample IV: Division into Cases and the Quotient-Remainder Theorem. Indirect Argument: Contradiction and Contraposition. Indirect Argument: Two Classical Theorems.
5. SEQUENCES, MATHEMATICAL INDUCTION, AND RECURSION.
Sequences. Mathematical Induction I. Mathematical Induction II. Strong Mathematical Induction and the Well-Ordering Principle. Defining Sequences Recursively. Solving Recurrence Relations by Iteration.
6. SET THEORY.
Set Theory: Definitions and the Element Method of Proof. Set Identities. Disproofs and Algebraic Proofs. Boolean Algebras and Russell's Paradox.
7. PROPERTIES OF FUNCTIONS.
Functions Defined on General Sets. One-to-one, Onto, and Inverse Functions. Composition of Functions. Cardinality and Sizes of Infinity.
8. PROPERTIES OF RELATIONS.
Relations on Sets. Reflexivity, Symmetry, and Transitivity. Equivalence Relations. Modular Arithmetic and Zn. The Euclidean Algorithm and Applications.
9. COUNTING.
Counting and Probability. The Multiplication Rule. Counting Elements of Disjoint Sets: The Addition Rule. The Pigeonhole Principle. Counting Subsets of a Set: Combinations. Pascal's Formula and the Binomial Theorem.
10. GRAPHS AND TREES.
Graphs: An Introduction. Trails, Paths, and Circuits. Matrix Representations of Graphs. Isomorphisms of Graphs. Trees: Examples and Basic Properties. Rooted Trees.