• 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 background better comprehend basic mathematical reasoning, and enables them to construct sound mathematical arguments.
• Around 2500 exercises providing ample practice for students, with numerous applied problems covering an impressive array of applications.
• Over 500 worked examples in problem-solution format. Proof solutions are intuitively developed in two steps, a discussion on how to approach the proof and a summary of the solution, to allow students the choice or quicker or more deliberate instruction depending on how well they understand the problem.
• 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.
• 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.
• A new Chapter 1 introduces students to some of the precise language that is a foundation for much mathematical thought, and is intended as a warm-up before addressing topics in more depth later on.
• New material on the definition of sound argument, trailing quantifiers, infinite unions and intersections, and Dijkstra''s shortest path algorithm.
• Expanded discussion of writing proofs and avoiding common mistakes in proof-writing.
• Increased coverage of functions of more than one variable and functions acting on sets.
• New margin notes are provided to provide better context-sensitive help highlighting issues of particular importance.

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. 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: 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. Indirect Argument: Two Classical Theorems. Application: Algorithms.

5. SEQUENCES, MATHEMATICAL INDUCTION, AND RECURSION.

Sequences. Mathematical Induction I. Mathematical Induction II. 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. FUNCTIONS.

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

8. 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. GRAPHS AND TREES.

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

11. ANALYZING ALGORITHM EFFICIENCY.

Real-Valued Functions of a Real Variable and Their Graphs. Application: Analysis of Algorithm Efficiency I. Exponential and Logarithmic Functions: Graphs and Orders. Application: Efficiency of Algorithms 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 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.