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| The Logic of Compound Statements | |
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| Logical Form and Logical Equivalence | |
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| Conditional Statements | |
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| Valid and Invalid Arguments | |
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| Application: Digital Logic Circuits | |
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| Application: Number Systems and Circuits for Addition | |
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| The Logic of Quantified Statements | |
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| Introduction to Predicates and Quantified Statements I | |
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| Introduction to Predicates and Quantified Statements II | |
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| Statements Containing Multiple Quantifiers | |
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| Arguments with Quantified Statements | |
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| Elementary Number Theory and Methods of Proof | |
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| Direct Proof and Counterexample I: Introduction | |
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| Direct Proof and Counterexample II: Rational Numbers | |
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| Direct Proof and Counterexample III: Divisibility | |
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| Direct Proof and Counterexample IV: Division into Cases and the Quotient-Remainder Theorem | |
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| Direct Proof and Counterexample V: Floor and Ceiling | |
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| Indirect Argument: Contradiction and Contraposition | |
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| Two Classical Theorems | |
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| Application: Algorithms | |
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| Sequences and Mathematical Induction | |
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| Sequences | |
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| Mathematical Induction I | |
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| Mathematical Induction II | |
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| Strong Mathematical Induction and the Well-Ordering Principle | |
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| Application: Correctness of Algorithms | |
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| Set Theory | |
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| Basic Definitions of Set Theory | |
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| Properties of Sets | |
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| Disproofs, Algebraic Proofs, and Boolean Algebras | |
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| Russell's Paradox and the Halting Problem | |
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| Counting and Probability | |
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| Introduction | |
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| Possibility Trees and the Multiplication Rule | |
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| Counting Elements of Disjoint Sets: The Addition Rule | |
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| Counting Subsets of a Set: Combinations | |
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| R-Combinations with Repetition Allowed | |
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| The Algebra of Combinations | |
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| The Binomial Theorem | |
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| Probability Axioms and Expected Value | |
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| Conditional Probability, Bayes' Formula, and Independent Events | |
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| Functions | |
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| Functions Defined on General Sets | |
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| One-to-One and Onto, Inverse Functions | |
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| Application: The Pigeonhole Principle | |
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| Composition of Functions | |
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| Cardinality with Applications to Computability | |
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| Recursion | |
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| Recursively Defined Sequences | |
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| Solving Recurrence Relations by Iteration | |
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| Second-Order Linear Homogeneous Recurrence Relations with Constant Coefficients | |
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| General Recursive Definitions | |
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| The Efficiency of Algorithms | |
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| Real-Valued Functions of a Real Variable and Their Graphs | |
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| O-, Omega-, and Theta-Notations | |
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| Application: Efficiency of Algorithms I | |
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| Exponential and Logarithmic Functions: Graphs and Orders | |
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| Application: Efficiency of Algorithms II | |
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| Relations | |
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| Relations on Sets | |
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| Reflexivity, Symmetry, and Transitivity | |
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| Equivalence Relations | |
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| Modular Arithmetic with Applications to Cryptography | |
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| Partial Order Relations | |
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| Graphs and Trees | |
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| Graphs: An Introduction | |
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| Paths and Circuits | |
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| Matrix Representations of Graphs | |
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| Isomorphisms of Graphs | |
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| Trees | |
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| Spanning Trees | |
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| Finite State Automata and Applications | |
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| Finite-State Automata | |
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| Application: Regular Expressions | |
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| Finite-State Automata | |
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| Simplifying Finite-State Automata | |
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| Appendices | |
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| Properties of the Real Numbers | |
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| Solutions and Hints to Selected Exercises | |