Course Summary
This course provides an introduction to the concept of mathematical proof and the techniques used in constructing and writing them. You’ll learn how to prove statements using mathematical induction, contradiction, and direct proofs.Key Learning Points
- Understand the importance of proofs in mathematics
- Learn common proof techniques such as mathematical induction and contradiction
- Develop skills in constructing and writing proofs
Related Topics for further study
Learning Outcomes
- Construct and write mathematical proofs
- Apply proof techniques to solve mathematical problems
- Understand the importance of proofs in mathematics
Prerequisites or good to have knowledge before taking this course
- Basic algebra skills
- Familiarity with mathematical notation
Course Difficulty Level
BeginnerCourse Format
- Self-paced
- Online
- Video Lectures
Similar Courses
- Introduction to Logic
- Discrete Mathematics
Related Education Paths
Notable People in This Field
- Terence Tao
- Eugenia Cheng
Related Books
Description
Mathematical thinking is crucial in all areas of computer science: algorithms, bioinformatics, computer graphics, data science, machine learning, etc. In this course, we will learn the most important tools used in discrete mathematics: induction, recursion, logic, invariants, examples, optimality. We will use these tools to answer typical programming questions like: How can we be certain a solution exists? Am I sure my program computes the optimal answer? Do each of these objects meet the given requirements?
Outline
- Making Convincing Arguments
- Promo Video
- Proofs?
- Proof by Example
- Impossibility Proof
- Impossibility Proof, II and Conclusion
- One Example is Enough
- Splitting an Octagon
- Making Fun in Real Life: Tensegrities (Optional)
- Know Your Rights
- Nobody Can Win All The Time: Nonexisting Examples
- Companion e-book
- Active Learning
- Python Programming Language
- Slides
- Slides
- Acknowledgements
- Puzzle: Tile a Chessboard
- Tiles, dominos, black and white, even and odd
- Puzzle: Two Congruent Parts
- Puzzle: Splitting
- How to Find an Example?
- Magic Squares
- Narrowing the Search
- Multiplicative Magic Squares
- More Puzzles
- Integer Linear Combinations
- Paths In a Graph
- Warm-up
- Subset without x and 100-x
- Rooks on a Chessboard
- Knights on a Chessboard
- Bishops on a Chessboard
- Subset without x and 2x
- N Queens: Brute Force Search
- N Queens: Backtracking: Example
- N Queens: Backtracking: Code
- 16 Diagonals
- Slides
- Slides
- N Queens: Brute Force Solution Code
- N Queens: Backtracking Solution Code
- 16 Diagonals: Code
- Slides
- Puzzle: Magic Square 3 times 3
- Puzzle: Different People Have Different Coins
- Puzzle: Free Accomodation
- Is there...
- Maximum Number of Two-digit Integers
- Maximum Number of Rooks on a Chessboard
- Maximum Number of Knights on a Chessboard
- Maximum Number of Bishops on a Chessboard
- Subset without x and 2x
- Puzzle: Maximum Number of Two Digit Integers
- Puzzle: N Queens
- Puzzle: 16 Diagonals
- Number of Solutions for the 8 Queens Puzzle
- Recursion and Induction
- Recursion
- Coin Problem
- Hanoi Towers
- Introduction, Lines and Triangles Problem
- Lines and Triangles: Proof by Induction
- Connecting Points
- Odd Points: Proof by Induction
- Sums of Numbers
- Bernoulli's Inequality
- Coins Problem
- Cutting a Triangle
- Flawed Induction Proofs
- Alternating Sum
- Two Cells of Opposite Colors: Hints
- Slides
- Slides
- Largest Amount that Cannot Be Paid with 5- and 7-Coins
- Pay Any Large Amount with 5- and 7-Coins (Optional)
- Puzzle: Hanoi Towers
- Puzzle: Two Cells of Opposite Colors
- Puzzle: Guess a Number
- Puzzle: Local Maximum (Optional)
- Puzzle: Connect Points
- Induction
- Logic
- Examples
- Counterexamples
- Basic Logic Constructs
- If-Then Generalization, Quantification
- Reductio ad Absurdum
- Balls in Boxes
- Numbers in Tables
- Pigeonhole Principle
- An (-1,0,1) Antimagic Square
- Handshakes
- Slides
- Slides
- Puzzle: Always Prime?
- Examples, Counterexamples and Logic
- Girls, Boys, and Two Languages
- Puzzle: Balls in Boxes
- Puzzle: Numbers in Boxes
- Puzzle: Numbers on the Chessboard
- Numbers in Boxes
- How to Pick Socks
- Pigeonhole Principle
- Puzzle: An (-1,0,1) Antimagic Square
- Invariants
- Double Counting
- `Homework Assignment' Problem
- Invariants
- More Coffee
- Debugging Problem
- Termination
- Arthur’s Books
- Even and Odd Numbers
- Summing up Digits
- Switching Signs
- Advanced Signs Switching
- Slides
- Slides
- Slides
- Slides
- Puzzle: Sums of Rows and Columns
- 'Homework Assignment' Problem
- 'Homework Assignment' Problem 2
- Girls and Boys
- Chess Tournaments
- Coffee with Milk
- More Coffee
- Debugging Problem
- Merging Bank Accounts
- Football Fans
- Puzzle: Arthur's Books
- Puzzle: Piece on a Chessboard
- Operations on Even and Odd Numbers
- Puzzle: Summing Up Digits
- Puzzle: Switching Signs
- Recolouring Chessboard
- Solving a 15-Puzzle
- The Rules of 15-Puzzle
- Permutations
- Proof: The Difficult Part
- Mission Impossible
- Classify a Permutation as Even/Odd
- Bonus Track: Fast Classification
- Project: The Task
- Quiz Hint: Why Every Even Permutation Is Solvable
- Reading
- Slides
- Even permutations
- Bonus Track: Finding The Sequence of Moves
- Puzzle: 15
- Transpositions and Permutations
- Neighbor transpositions
- Is a permutation even?
- Bonus Track: Algorithm for 15-Puzzle
Summary of User Reviews
This course on What is a Proof has received positive reviews from learners. Many users appreciate the engaging and clear presentation of the material that makes it easy to understand.Key Aspect Users Liked About This Course
The course effectively breaks down complex mathematical proofs into manageable sections, making it easier for learners to follow and comprehend.Pros from User Reviews
- Clear and engaging presentation of the material
- Effective breakdown of complex proofs into manageable sections
- Use of real-world examples to illustrate concepts
- Good pacing and structure of the course
- Challenging exercises that help reinforce learning
Cons from User Reviews
- Some learners found the course to be too basic
- The course may not be suitable for advanced learners looking for more in-depth information
- Lack of interaction with instructors and other learners
- Limited practical applications for the material covered
- Some learners reported technical issues with the platform
Alexander S. Kulikov
4.4 Raiting
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