Higher Education

Author(s): Susanna S. Epp

ISBN: 9789355731029

5th Edition

Copyright: 2020

India Release: 2021

Binding: Paperback

Pages: 1056

Trim Size: 254 x 203 mm

DISCRETE MATHEMATICS WITH APPLICATIONS, 5th Edition, Metric Edition explains complex, abstract concepts with clarity and precision and provides a strong foundation for computer science and upper-level mathematics courses of the computer age. Author Susanna Epp presents not only the major themes of discrete mathematics, but also the reasoning that underlies mathematical thought. Students develop the ability to think abstractly as they study the ideas of logic and proof. While learning about such concepts as logic circuits and computer addition, algorithm analysis, recursive thinking, computability, automata, cryptography and combinatorics, students discover that the ideas of discrete mathematics underlie and are essential to today's science and technology.

- More than 2500 exercises provide ample practice for students, with numerous applied problems covering an impressive array of applications.
- Epp addresses inherent difficulties in understanding logic and language with very concrete and easy-to-conceptualize examples, an approach that helps students with a variety of backgrounds better comprehend basic mathematical reasoning, and enables them to construct sound mathematical arguments.
- Features, definitions, theorems, and exercise types are clearly marked and easily navigable, making the book an excellent reference that students will want to keep and continually refer back to in their later courses.
- Flexible organization, allowing instructors the ability to mix core and optional topics easily to suit a wide variety of discrete math course syllabi and topic focus.
- Over 500 worked examples in problem-solution format guide students in building a conceptual understanding of how to solve problems. Proof solutions are intuitively developed in two steps, a discussion on how to approach the proof and a summary of the solution, allowing students the choice of faster or more deliberate instruction depending on how well they understand the problem.
- A large number of computer science applications are included, both to motivate students and to ease their transition into more advanced computer science courses.
- New material was added on binary search trees, cryptographic hash functions, bound variables and scope in mathematics and computer programming, and the use of cryptography for message authentication.
- Sections on quantifier use, the concept of mathematical proof, proof-writing advice, and set theory proofs were revised and expanded based on classroom experience, and new types of exercises were added.
- The sections introducing mathematical induction were reorganized, discussion of structural induction was expanded, and additional applications of induction were provided.
- Discussion of strings and graphs begins in Chapter 1 and is integrated with applications throughout the text. The handshake theorem, previously in Chapter 12, is now in Chapter 4.
- Explanations for two’s complements and for O-, Ω-, and Θ-Notations were significantly simplified.

1. SPEAKING MATHEMATICALLY.

Variables. The Language of Sets. The Language of Relations and Functions. The Language of Graphs.

2. THE LOGIC OF COMPOUND STATEMENTS.

Logical Form and Logical Equivalence.

Conditional Statements.

Valid and Invalid Arguments.

Application: Digital Logic Circuits.

Application: Number Systems and Circuits for Addition.

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: Writing Advice. Direct Proof and Counterexample III: Rational Numbers. Direct Proof and Counterexample IV: Divisibility. Direct Proof and Counterexample V: Division into Cases and the Quotient-Remainder Theorem. Direct Proof and Counterexample VI: Floor and Ceiling. Indirect Argument: Contradiction and Contraposition. Indirect Argument: Two Famous Theorems. Application: Algorithms.

5. SEQUENCES, MATHEMATICAL INDUCTION, AND RECURSION.

Sequences. Mathematical Induction I: Proving Formulas. Mathematical Induction II: Applications. Strong Mathematical Induction and the Well-Ordering Principle. Application: Correctness of Algorithms. Defining Sequences Recursively. Solving Recurrence Relations by Iteration. Second-Order Linear Homogeneous Recurrence Relations with Constant Coefficients. General Recursive Definitions and Structural Induction.

6. SET THEORY.

Set Theory: Definitions and the Element Method of Proof. Properties of Sets. Disproofs and Algebraic Proofs. Boolean Algebras, Russell’s Paradox, and the Halting Problem.

7. PROPERTIES OF FUNCTIONS.

Functions Defined on General Sets. One-to-one, Onto, and Inverse Functions. Composition of Functions. Cardinality with Applications to Computability.

8. PROPERTIES OF RELATIONS.

Relations on Sets. Reflexivity, Symmetry, and Transitivity. Equivalence Relations. Modular Arithmetic with Applications to Cryptography. Partial Order Relations.

9. COUNTING AND PROBABILITY

Introduction. Possibility Trees and the Multiplication Rule. Counting Elements of Disjoint Sets: The Addition Rule. The Pigeonhole Principle. Counting Subsets of a Set: Combinations. r-Combinations with Repetition Allowed. Pascal’s Formula and the Binomial Theorem. Probability Axioms and Expected Value. Conditional Probability, Bayes’ Formula, and Independent Events.

10. THEORY OF GRAPHS AND TREES.

Trails, Paths, and Circuits. Matrix Representations of Graphs. Isomorphisms of Graphs. Trees: Examples and Basic Properties. Rooted Trees. Spanning Trees and a Shortest Path Algorithm.

11. ANALYSIS OF ALGORITHM EFFICIENCY.

Real-Valued Functions of a Real Variable and Their Graphs. O-, Ω-, and Θ-Notations. Application: Analysis of Algorithm Efficiency I. Exponential and Logarithmic Functions: Graphs and Orders. Application: Analysis of Algorithm Efficiency II.

12. REGULAR EXPRESSIONS AND FINITE STATE AUTOMATA.

Formal Languages and Regular Expressions. Finite-State Automata. Simplifying Finite-State Automata.

**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 Emerita 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 published a number of articles related to this topic, one of which was chosen for inclusion in The Best Writing on Mathematics 2012. She has spoken widely on discrete mathematics and 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. The third edition of Discrete Mathematics with Applications received a Texty Award for Textbook Excellence in June 2005. 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. She received the Hay Award for Contributions to Mathematics Education in 2005 and the Award for Distinguished Teaching given by the Illinois Section of the MAA in 2010.