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Speaking Mathematically | |
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Variables | |
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The Language of Sets | |
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The Language of Relations and Functions | |
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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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Predicates and Quantified Statements I | |
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Predicatesand Quantified Statements II | |
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Statements with 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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Indirect Argument: Two Classical Theorems | |
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Application: Algorithms | |
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Sequences, Mathematical Induction, and Recursion | |
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Sequences | |
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Mathematical Induction I | |
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MathematicalInduction 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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Defining Sequences Recursively | |
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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 and Structural Induction | |
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Set Theory | |
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Set Theory: Definitions and the Element Method of Proof | |
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Set Identities | |
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Disproofs and Algebraic Proofs | |
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Boolean Algebras, Russell's Paradox, and the Halting Problem | |
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Properties of Functions | |
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Functions Defined on General Sets | |
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One-to-one, Onto, Inverse Functions | |
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Composition of Functions | |
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Cardinality, Sizes of Infinity, and Applications to Computability | |
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Properties of Relations | |
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Relations on Sets (add material about relational databases) | |
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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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Counting | |
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Counting and Probability | |
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The Multiplication Rule | |
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Counting Elements of Disjoint Sets: The Addition Rule | |
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The Pigeonhole Principle | |
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Counting Subsets of a Set: Combinations. r-Combinations with Repetition Allowed | |
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Pascal's Formula and 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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Graphs and Trees | |
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Graphs: An Introduction | |
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Trails, Paths, and Circuits | |
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Matrix Representations of Graphs | |
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Isomorphisms of Graphs | |
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Trees: Examples and Basic Properties | |
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Rooted Trees | |
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Spanning Trees and a Shortest Path Algorithm | |
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Analyzing Algorithm Efficiency | |
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Real-Valued Functions of a Real Variable and Their Graphs | |
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O-, ?-, and ?-Notations | |
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Application: Efficiency of Algorithms I. Exponential and Logarithmic Functions: Graphs and Orders | |
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Application: Efficiency of Algorithms II | |
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Regular Expressions and Finite State Automata | |
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Formal Languages and Regular Expressions | |
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Finite-State Automata | |
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Simplifying Finite-State Automata | |