Speaking Mathematically | |

Variables | |

The Language of Sets | |

The Language of Relations and Functions | |

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 | |

The Logic of Quantified Statements | |

Predicates and Quantified Statements I | |

Predicatesand Quantified Statements II | |

Statements with Multiple Quantifiers | |

Arguments with Quantified Statements | |

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 | |

Sequences, Mathematical Induction, and Recursion | |

Sequences | |

Mathematical Induction I | |

MathematicalInduction 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 | |

Set Theory | |

Set Theory: Definitions and the Element Method of Proof | |

Set Identities | |

Disproofs and Algebraic Proofs | |

Boolean Algebras, Russell's Paradox, and the Halting Problem | |

Properties of Functions | |

Functions Defined on General Sets | |

One-to-one, Onto, Inverse Functions | |

Composition of Functions | |

Cardinality, Sizes of Infinity, and Applications to Computability | |

Properties of Relations | |

Relations on Sets (add material about relational databases) | |

Reflexivity, Symmetry, and Transitivity | |

Equivalence Relations | |

Modular Arithmetic with Applications to Cryptography | |

Partial Order Relations | |

Counting | |

Counting and Probability | |

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 | |

Graphs and Trees | |

Graphs: An Introduction | |

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 | |

Analyzing Algorithm Efficiency | |

Real-Valued Functions of a Real Variable and Their Graphs | |

O-, ?-, and ?-Notations | |

Application: Efficiency of Algorithms I. Exponential and Logarithmic Functions: Graphs and Orders | |

Application: Efficiency of Algorithms II | |

Regular Expressions and Finite State Automata | |

Formal Languages and Regular Expressions | |

Finite-State Automata | |

Simplifying Finite-State Automata | |

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