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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. 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: 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.
5. SEQUENCES, MATHEMATICAL INDUCTION, AND RECURSION. Sequences. Mathematical Induction I. Mathematical Induction 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.
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. FUNCTIONS.Functions Defined on General Sets. One-to-one, Onto, Inverse Functions. Composition of Functions. Cardinalitywith Applications to Computability.
8. 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. GRAPHS AND TREES. Graphs: Definitions and Basic Properties. Trails, Paths, and Circuits. Matrix Representations of Graphs. Isomorphisms of Graphs. Trees. Rooted Trees. Spanning Trees and a Shortest Path Algorithm.11. ANALYZING ALGORITHM EFFICIENCY.Real-Valued Functions of a Real Variable and Their Graphs. Application: Analysis of Algorithm Efficiency I. Exponential and Logarithmic Functions: Graphs and Orders. Application: Efficiency of Algorithms II.
12. REGULAR EXPRESSIONS AND FINITE STATE AUTOMATA.Formal Languages and Regular Expressions. Finite-State Automata. Simplifying Finite-State Automata.