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
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.
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.
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.