| Speaking Mathematically | |
| Variables | |
| The Language of Sets | |
| The Language of Relations and Functions | |
| 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 | |
| The Logic of Quantified Statements | |
| Predicates and Quantified Statements I | |
| Predicatesand Quantified Statements II | |
| Statements with Multiple Quantifiers | |
| Arguments with Quantified Statements | |
| 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 | |
| Sequences, Mathematical Induction, and Recursion | |
| Sequences | |
| Mathematical Induction I | |
| MathematicalInduction 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 | |
| Set Theory | |
| Set Theory: Definitions and the Element Method of Proof | |
| Set Identities | |
| Disproofs and Algebraic Proofs | |
| Boolean Algebras, Russell's Paradox, and the Halting Problem | |
| Properties of Functions | |
| Functions Defined on General Sets | |
| One-to-one, Onto, Inverse Functions | |
| Composition of Functions | |
| Cardinality, Sizes of Infinity, and Applications to Computability | |
| Properties of Relations | |
| Relations on Sets (add material about relational databases) | |
| Reflexivity, Symmetry, and Transitivity | |
| Equivalence Relations | |
| Modular Arithmetic with Applications to Cryptography | |
| Partial Order Relations | |
| Counting | |
| Counting and Probability | |
| 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 | |
| Graphs and Trees | |
| Graphs: An Introduction | |
| 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 | |
| Analyzing Algorithm Efficiency | |
| Real-Valued Functions of a Real Variable and Their Graphs | |
| O-, ?-, and ?-Notations | |
| Application: Efficiency of Algorithms I. Exponential and Logarithmic Functions: Graphs and Orders | |
| Application: Efficiency of Algorithms II | |
| Regular Expressions and Finite State Automata | |
| Formal Languages and Regular Expressions | |
| Finite-State Automata | |
| Simplifying Finite-State Automata | |
| Table of Contents provided by Publisher. All Rights Reserved. |
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