**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.