1. SPEAKING MATHEMATICALLY.

Variables. The Language of Sets. The Language of Relations and Functions. The Language of Graphs.

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: Writing Advice. Direct Proof and Counterexample III: Rational Numbers. Direct Proof and Counterexample IV: Divisibility. Direct Proof and Counterexample V: Division into Cases and the Quotient-Remainder Theorem. Direct Proof and Counterexample VI: Floor and Ceiling. Indirect Argument: Contradiction and Contraposition. Indirect Argument: Two Famous Theorems. Application: Algorithms.

5. SEQUENCES, MATHEMATICAL INDUCTION, AND RECURSION.
Sequences. Mathematical Induction I: Proving Formulas. Mathematical Induction II: Applications. 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. PROPERTIES OF FUNCTIONS.
Functions Defined on General Sets. One-to-one, Onto, and Inverse Functions. Composition of Functions. Cardinality with Applications to Computability.

8. PROPERTIES OF 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. THEORY OF GRAPHS AND TREES.
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.

11. ANALYSIS OF ALGORITHM EFFICIENCY.
Real-Valued Functions of a Real Variable and Their Graphs. O-, -, and -Notations. Application: Analysis of Algorithm Efficiency I. Exponential and Logarithmic Functions: Graphs and Orders. Application: Analysis of Algorithm Efficiency II.

12. REGULAR EXPRESSIONS AND FINITE STATE AUTOMATA.
Formal Languages and Regular Expressions. Finite-State Automata. Simplifying Finite-State Automata.