فهرست مطالب

• About the Author
• About the Technical Reviewer
• Contents in Detail
• Foreword
• Acknowledgments
• Introduction
• 1. Computers and Numbers
• Numbers and Number Bases
• How Computers Represent Numbers
• What You Need to Know About Floating-Point Arithmetic
• Summary
• 2. Sets and Abstract Algebra
• Concerning Sets
• The Laws of Set Theory
• Experimenting with Number Sets in Python
• Abstract Algebra and Groups
• Cantor: To Infinity and Beyond
• Computable and Uncomputable Numbers
• Uncomputable Python
• Summary
• 3. Boolean Algebra
• Definition and Laws
• Boolean Algebra Leads to Digital Logic
• Boolean Functions
• Canonical Normal Forms
• Karnaugh Maps
• Algebraic Simplification of Boolean Expressions
• Digital Circuits
• Summary
• 4. Functions and Relations
• Functions
• Binary Relations
• Equivalence Relations and Classes
• Partial Orderings
• Summary
• 5. Induction
• Weak Induction
• Strong Induction
• Cautionary Tales
• Proving with Loop Invariants
• Summary
• 6. Recurrence and Recursion
• Recurrence Relations
• Nonlinear Recurrence Relations
• Solving Recurrence Relations
• Recursion
• Summary
• 7. Number Theory
• Primes
• Divisibility
• Modular Arithmetic
• Diophantine Equations
• Integer Sequences
• Summary
• 8. Counting and Combinatorics
• The Principles of Counting
• An Example: Hats in the Park
• Sum and Product Rules
• Inclusion-Exclusion
• The Pigeonhole Principle
• Permutations and Combinations
• The Binomial Theorem
• Summary
• 9. Graphs
• Basic Graph Concepts
• Representing Graphs in Code
• Breadth-First and Depth-First Traversal and Searching
• The Shortest Path Between Nodes
• Directed Acyclic Graphs and Topological Sort
• Summary
• 10. Trees
• Defining Trees
• Spanning Trees
• Rooted and Ordered Trees
• Binary Trees
• Binary Tree Traversals and Searches
• Binary Trees in Code
• The Animals Game
• Summary
• 11. Probability
• Events and Random Variables
• The Rules of Probability
• Joint and Marginal Probabilities
• Probability Distributions
• The Central Limit Theorem and the Law of Large Numbers
• Bayes’ Theorem
• Summary
• 12. Statistics
• Types of Data
• Summary Statistics
• Quantiles and Box Plots
• Correlation
• A Cautionary Tale: Anscombe’s Quartet
• Hypothesis Testing
• Confidence Intervals
• Summary
• 13. Linear Algebra
• Vectors and Vector Spaces
• Vector Space Concepts
• Matrices
• Solving Systems of Linear Equations
• Square Matrices
• Linear Transformations
• Eigenvalues and Eigenvectors
• Summary
• 14. Differential Calculus
• Derivatives
• Minima and Maxima of Functions
• Partial Derivatives
• Derivatives in Code
• Optimization with Gradient Descent
• Summary
• 15. Integral Calculus
• Curves and Areas
• Indefinite Integrals
• Definite Integrals
• Summary
• 16. Differential Equations
• Ordinary Differential Equations
• Solving ODEs Numerically
• Worked Examples
• Summary
• Final Words
• Index