Discrete Math with Proof

Summary

For one-semester undergraduate Discrete Mathematics courses for math and computer science majors.

This exceedingly well-written text equips students to apply discrete mathematics and provides opportunities for practice of the concepts presented. A semester of Calculus is useful for reasons of sophistication.

Table of Contents

1. Introduction.

What Is Discrete Mathematics? The Stable Marriage Problem. Other Examples. Exercises.

2. Sets, Logic, and Boolean Algebras.

Sets. Logic in Daily Life. Propositional Logic. A Detailed Look at Tautologies. Boolean Algebras. Predicate Logic. Analyzing Claims (Optional). Quick Check Solutions. Projects.

3. Proof.

Introduction to Mathematical Proof. Proof Strategies. Mathematical Induction. Creating Proofs: Hints and Suggestions. Quick Check Solutions.

4. Algorithms.

Expressing Algorithms. Measuring Algorithm Efficiency. Pattern Matching. The Halting Problem. Quick Check Solutions.

5. Counting.

Permutations and Combinations. Combinatorial Proofs. Pigeon-Hole: Inclusion-Exclusion. Quick Check Solutions.

6. Finite Probability Theory.

The Language of Probabilities. Conditional Probabilities and Independent Events. Counting and Probability. Expected Value. Bayes' Theorem. Quick Check Solutions.

7. Recursion.

Recursive Algorithms. Recurrence Relations. Generating Functions. The Josephus Problem. Quick Check Solutions. Projects.

8. Combinatorics.

Partitions, Occupancy Problems, and Stirling Numbers. Latin Squares; Finite Projective Planes. Balanced Incomplete Block Designs. The Knapsack Problem. Error-Correcting Codes. Distinct Representatives; Ramsey Numbers. Quick Check Solutions.

9. Formal Models in Computer Science.

Information. Finite-State Machines. Formal Languages. Regular Expressions. The Three Faces of Regular. A Glimpse at More Advanced Topics. Quick Check Solutions.

10. Graphs.

Terminology. Connectivity and Adjacency. Euler and Hamilton. Representation and Isomorphism. The Big Theorems: Planarity, Polyhedra, Chromatic Number. Directed Graphs and Weighted Graphs. Quick Check Solutions.

11. Trees.

Terminology, Counting. Traversal, Searching, and Sorting. More Applications of Trees. Spanning Trees. Quick Check Solutions.

12. Functions, Relations, Databases, and Circuits.

Functions and Relations. Equivalence Relations. n-ary Relations and Relational Databases. Binary Functions and Binary Expressions. Combinatorial Circuits. Quick Check Solutions.

Appendix A. Number Systems.

The Natural Numbers. The Integers. The Rational Numbers. The Real Numbers. The Complex Numbers. Other Number Systems. Representation of Numbers.

Appendix B. Summation Notation.
Appendix C. Logic Puzzles.

Logic Puzzles about AND, OR, NOT. Logic Puzzles about Implication, Biconditional, and Equivalence. Exercises. Quick Check Solutions.

Appendix D. The Golden Ratio.
Appendix E. Matrices.
References.
Index.