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Субтитры: Английский

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Mathematical InductionProof TheoryDiscrete MathematicsMathematical Logic

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Субтитры: Английский

Программа курса: что вы изучите

3 ч. на завершение

Making Convincing Arguments

Why some arguments are convincing and some are not? What makes an argument convincing? How to establish your argument in such a way that there is no possible room for doubt left? How mathematical thinking can help with this? In this week we will start digging into these questions. We will see how a small remark or a simple observation can turn a seemingly non-trivial question into an obvious one. Through various examples we will observe a parallel between constructing a rigorous argument and mathematical reasoning.

10 видео ((всего 43 мин.)), 4 материалов для самостоятельного изучения, 4 тестов
10 видео
Proof by Example1мин
Impossibility Proof2мин
Impossibility Proof, II and Conclusion3мин
One Example is Enough3мин
Splitting an Octagon1мин
Making Fun in Real Life: Tensegrities10мин
Know Your Rights5мин
Nobody Can Win All The Time: Nonexisting Examples8мин
4 материала для самостоятельного изучения
1 практическое упражнение
Tiles, dominos, black and white, even and odd6мин
5 ч. на завершение

How to Find an Example?

How can we be certain that an object with certain requirements exist? One way to show this, is to go through all objects and check whether at least one of them meets the requirements. However, in many cases, the search space is enormous. A computer may help, but some reasoning that narrows the search space is important both for computer search and for "bare hands" work. In this module, we will learn various techniques for showing that an object exists and that an object is optimal among all other objects. As usual, we'll practice solving many interactive puzzles. We'll show also some computer programs that help us to construct an example.

16 видео ((всего 90 мин.)), 6 материалов для самостоятельного изучения, 12 тестов
16 видео
Narrowing the Search6мин
Multiplicative Magic Squares5мин
More Puzzles9мин
Integer Linear Combinations5мин
Paths In a Graph4мин
N Queens: Brute Force Search (Optional)10мин
N Queens: Backtracking: Example (Optional)7мин
N Queens: Backtracking: Code (Optional)7мин
16 Diagonals (Optional)3мин
Subset without x and 100-x4мин
Rooks on a Chessboard2мин
Knights on a Chessboard5мин
Bishops on a Chessboard2мин
Subset without x and 2x6мин
6 материала для самостоятельного изучения
N Queens: Brute Force Solution Code (Optional)10мин
N Queens: Backtracking Solution Code (Optional)10мин
16 Diagonals: Code (Optional)10мин
Slides (Optional)1мин
3 практического упражнения
Is there...20мин
Number of Solutions for the 8 Queens Puzzle (Optional)20мин
Maximum Number of Two-digit Integers2мин
6 ч. на завершение

Recursion and Induction

We'll discover two powerful methods of defining objects, proving concepts, and implementing programs — recursion and induction. These two methods are heavily used, in particular, in algorithms — for analysing correctness and running time of algorithms as well as for implementing efficient solutions. You will see that induction is as simple as falling dominos, but allows to make convincing arguments for arbitrarily large and complex problems by decomposing them and moving step by step. You will learn how famous Gauss unexpectedly solved his teacher's problem intended to keep him busy the whole lesson in just two minutes, and in the end you will be able to prove his formula using induction. You will be able to generalize scary arithmetic exercises and then solve them easily using induction.

13 видео ((всего 111 мин.)), 3 материалов для самостоятельного изучения, 8 тестов
13 видео
Coin Problem4мин
Hanoi Towers7мин
Introduction, Lines and Triangles Problem10мин
Lines and Triangles: Proof by Induction5мин
Connecting Points12мин
Odd Points: Proof by Induction5мин
Sums of Numbers8мин
Bernoulli's Inequality8мин
Coins Problem9мин
Cutting a Triangle8мин
Flawed Induction Proofs9мин
Alternating Sum9мин
3 материала для самостоятельного изучения
Two Cells of Opposite Colors: Hints10мин
5 практического упражнения
Largest Amount that Cannot Be Paid with 5- and 7-Coins10мин
Pay Any Large Amount with 5- and 7-Coins20мин
Number of Moves to Solve the Hanoi Towers Puzzle30мин
Two Cells of Opposite Colors: Feedback
3 ч. на завершение


We have already invoked mathematical logic when we discussed how to make convincing arguments by giving examples. This week we will turn mathematical logic full on. We will discuss its basic operations and rules. We will see how logic can play a crucial and indispensable role in creating convincing arguments. We will discuss how to construct a negation to the statement, and you will see how to win an argument by showing your opponent is wrong with just one example called counterexample!. We will see tricky and seemingly counterintuitive, but yet (an unintentional pun) logical aspects of mathematical logic. We will see one of the oldest approaches to making convincing arguments: Reductio ad Absurdum.

10 видео ((всего 53 мин.)), 2 материалов для самостоятельного изучения, 9 тестов
10 видео
Basic Logic Constructs10мин
If-Then Generalization, Quantification8мин
Reductio ad Absurdum4мин
Balls in Boxes4мин
Numbers in Tables5мин
Pigeonhole Principle2мин
An (-1,0,1) Antimagic Square2мин
2 материала для самостоятельного изучения
4 практического упражнения
Examples, Counterexamples and Logic14мин
Numbers in Boxes5мин
How to Pick Socks5мин
Pigeonhole Principle10мин
Рецензии: 112Chevron Right


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получил значимые преимущества в карьере благодаря этому курсу

Лучшие отзывы о курсе Mathematical Thinking in Computer Science

автор: ADMar 26th 2019

The teachers are informative and good. They explain the topic in a way that we can easily understand. The slides provide all the information that is needed. The external tools are fun and informative.

автор: JVOct 16th 2017

I really liked this course, it's a good introduction to mathematical thinking, with plenty of examples and exercises, I also liked the use of other external graphical tools as exercises.



Alexander S. Kulikov

Visiting Professor
Department of Computer Science and Engineering

Michael Levin

Computer Science

Vladimir Podolskii

Associate Professor
Computer Science Department

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