Discrete Mathematics with Applications, 3rd Edition
- Susanna S. Epp DePaul University
- ISBN-10: 0534359450 | ISBN-13: 9780534359454
- 928 Pages
- Previous Editions: 1995, 1990
- © 2004 | Published
- Newer Edition Available
Susanna Epp’s DISCRETE MATHEMATICS, THIRD EDITION provides a clear introduction to discrete mathematics. Renowned for her lucid, accessible prose, Epp explains complex, abstract concepts with clarity and precision. This book 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 the science and technology of the computer age. Overall, Epp’s emphasis on reasoning provides students with a strong foundation for computer science and upper-level mathematics courses.
1. 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.
2. THE LOGIC OF QUANTIFIED STATEMENTS.
Introduction to Predicates and Quantified Statements I. Introduction to Predicates and Quantified Statements II. Statements Containing Multiple Quantifiers. Arguments with Quantified Statements.
3. 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. Direct Proof and Counterexample V: Floor and Ceiling. Indirect Argument: Contradiction and Contraposition. Two Classical Theorems. Application: Algorithms.
4. SEQUENCES AND MATHEMATICAL INDUCTION.
Sequences. Mathematical Induction I. Mathematical Induction II. Strong Mathematical Induction and the Well-Ordering Principle. Application: Correctness of Algorithms.
5. SET THEORY.
Basic Definitions of Set Theory. Properties of Sets. Disproofs, Algebraic Proofs, and Boolean Algebras. Russell’s Paradox and the Halting Problem.
6. COUNTING AND PROBABILITY.
Introduction. Possibility Trees and the Multiplication Rule. Counting Elements of Disjoint Sets: The Addition Rule. Counting Subsets of a Set: Combinations. R-Combinations with Repetition Allowed. The Algebra of Combinations. The Binomial Theorem. Probability Axioms and Expected Value. Conditional Probability, Bayes’ Formula, and Independent Events.
7. FUNCTIONS.
Functions Defined on General Sets. One-to-One and Onto, Inverse Functions. Application: The Pigeonhole Principle. Composition of Functions. Cardinality with Applications to Computability.
8. RECURSION.
Recursively Defined Sequences. Solving Recurrence Relations by Iteration. Second-Order Linear Homogeneous Recurrence Relations with Constant Coefficients. General Recursive Definitions.
9. THE EFFICIENCY OF ALGORITHMS.
Real-Valued Functions of a Real Variable and Their Graphs. O-, Omega-, and Theta-Notations. Application: Efficiency of Algorithms I. Exponential and Logarithmic Functions: Graphs and Orders. Application: Efficiency of Algorithms II.
10. RELATIONS.
Relations on Sets. Reflexivity, Symmetry, and Transitivity. Equivalence Relations. Modular Arithmetic with Applications to Cryptography. Partial Order Relations.
11. GRAPHS AND TREES.
Graphs: An Introduction. Paths and Circuits. Matrix Representations of Graphs. Isomorphisms of Graphs. Trees. Spanning Trees.
12. FINITE STATE AUTOMATA AND APPLICATIONS.
Finite-State Automata. Application: Regular Expressions. Finite-State Automata. Simplifying Finite-State Automata.
Appendices.
Properties of the Real Numbers. Solutions and Hints to Selected Exercises.
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Provides additional explanations for some topics and complete, worked-out solutions to every third problem in the text that is not fully solved in Appendix B.
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