1.1 · Variables
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TOPICS AND PRACTICE
68 step-by-step lessons and 205 practice exercises.
68 step-by-step lessons205 graded practice tasks
Each lesson provides a structured foundation for its topic; it is not a full textbook translation.
Canvas course assignments are available separately from the trainer.
1Speaking Mathematically4 sections
1.2 · The Language of Sets
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1.3 · The Language of Relations and Functions
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1.4 · The Language of Graphs
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2The Logic of Compound Statements5 sections
2.1 · Logical Form and Logical Equivalence
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2.2 · Conditional Statements
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2.3 · Valid and Invalid Arguments
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2.4 · Application: Digital Logic Circuits
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2.5 · Application: Number Systems and Circuits for Addition
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3The Logic of Quantified Statements4 sections
3.1 · Predicates and Quantified Statements I
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3.2 · Predicates and Quantified Statements II
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3.3 · Statements with Multiple Quantifiers
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3.4 · Arguments with Quantified Statements
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4Elementary Number Theory and Methods of Proof10 sections
4.1 · Direct Proof and Counterexample I: Introduction
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4.2 · Direct Proof and Counterexample II: Writing Advice
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4.3 · Direct Proof and Counterexample III: Rational Numbers
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4.4 · Direct Proof and Counterexample IV: Divisibility
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4.5 · Direct Proof and Counterexample V: Division into Cases and the Quotient-Remainder Theorem
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4.6 · Direct Proof and Counterexample VI: Floor and Ceiling
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4.7 · Indirect Argument: Contradiction and Contraposition
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4.8 · Indirect Argument: Two Famous Theorems
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4.9 · Application: The Handshake Theorem
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4.10 · Application: Algorithms
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5Sequences, Mathematical Induction, and Recursion9 sections
5.1 · Sequences
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5.2 · Mathematical Induction I: Proving Formulas
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5.3 · Mathematical Induction II: Applications
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5.4 · Strong Mathematical Induction and the Well-Ordering Principle for the Integers
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5.5 · Application: Correctness of Algorithms
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5.6 · Defining Sequences Recursively
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5.7 · Solving Recurrence Relations by Iteration
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5.8 · Second-Order Linear Homogeneous Recurrence Relations with Constant Coefficients
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5.9 · General Recursive Definitions and Structural Induction
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6Set Theory4 sections
6.1 · Set Theory: Definitions and the Element Method of Proof
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6.2 · Properties of Sets
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6.3 · Disproofs and Algebraic Proofs
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6.4 · Boolean Algebras, Russell’s Paradox, and the Halting Problem
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7Properties of Functions4 sections
7.1 · Functions Defined on General Sets
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7.2 · One-to-One, Onto, and Inverse Functions
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7.3 · Composition of Functions
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7.4 · Cardinality with Applications to Computability
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8Properties of Relations5 sections
8.1 · Relations on Sets
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8.2 · Reflexivity, Symmetry, and Transitivity
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8.3 · Equivalence Relations
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8.4 · Modular Arithmetic with Applications to Cryptography
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8.5 · Partial Order Relations
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9Counting and Probability9 sections
9.1 · Introduction to Probability
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9.2 · Possibility Trees and the Multiplication Rule
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9.3 · Counting Elements of Disjoint Sets: The Addition Rule
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9.4 · The Pigeonhole Principle
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9.5 · Counting Subsets of a Set: Combinations
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9.6 · r-Combinations with Repetition Allowed
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9.7 · Pascal’s Formula and the Binomial Theorem
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9.8 · Probability Axioms and Expected Value
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9.9 · Conditional Probability, Bayes’ Formula, and Independent Events
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10Theory of Graphs and Trees6 sections
10.1 · Trails, Paths, and Circuits
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10.2 · Matrix Representations of Graphs
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10.3 · Isomorphisms of Graphs
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10.4 · Trees: Examples and Basic Properties
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10.5 · Rooted Trees
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10.6 · Spanning Trees and a Shortest Path Algorithm
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11Analysis of Algorithm Efficiency5 sections
11.1 · Real-Valued Functions of a Real Variable and Their Graphs
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11.2 · Big-O, Big-Omega, and Big-Theta Notations
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11.3 · Application: Analysis of Algorithm Efficiency I
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11.4 · Exponential and Logarithmic Functions: Graphs and Orders
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11.5 · Application: Analysis of Algorithm Efficiency II
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12Regular Expressions and Finite-State Automata3 sections
12.1 · Formal Languages and Regular Expressions
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12.2 · Finite-State Automata
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12.3 · Simplifying Finite-State Automata
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Appendix B contains selected answers and hints, not solutions to every exercise.