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初級レベル

英語

字幕:英語

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Mathematical InductionProof TheoryDiscrete MathematicsMathematical Logic

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初級レベル

英語

字幕:英語

シラバス - 本コースの学習内容

1
3時間で修了

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件のビデオ (合計43分), 4 readings, 4 quizzes
10件のビデオ
Proofs?3 分
Proof by Example1 分
Impossibility Proof2 分
Impossibility Proof, II and Conclusion3 分
One Example is Enough3 分
Splitting an Octagon1 分
Making Fun in Real Life: Tensegrities10 分
Know Your Rights5 分
Nobody Can Win All The Time: Nonexisting Examples8 分
4件の学習用教材
Slides10 分
Python10 分
Slides1 分
Acknowledgements1 分
1の練習問題
Tiles, dominos, black and white, even and odd6 分
2
5時間で修了

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件のビデオ (合計90分), 6 readings, 12 quizzes
16件のビデオ
Narrowing the Search6 分
Multiplicative Magic Squares5 分
More Puzzles9 分
Integer Linear Combinations5 分
Paths In a Graph4 分
N Queens: Brute Force Search (Optional)10 分
N Queens: Backtracking: Example (Optional)7 分
N Queens: Backtracking: Code (Optional)7 分
16 Diagonals (Optional)3 分
Warm-up5 分
Subset without x and 100-x4 分
Rooks on a Chessboard2 分
Knights on a Chessboard5 分
Bishops on a Chessboard2 分
Subset without x and 2x6 分
6件の学習用教材
Slides1 分
N Queens: Brute Force Solution Code (Optional)10 分
N Queens: Backtracking Solution Code (Optional)10 分
16 Diagonals: Code (Optional)10 分
Slides (Optional)1 分
Slides1 分
3の練習問題
Is there...20 分
Number of Solutions for the 8 Queens Puzzle (Optional)20 分
Maximum Number of Two-digit Integers2 分
3
6時間で修了

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件のビデオ (合計111分), 3 readings, 8 quizzes
13件のビデオ
Coin Problem4 分
Hanoi Towers7 分
Introduction, Lines and Triangles Problem10 分
Lines and Triangles: Proof by Induction5 分
Connecting Points12 分
Odd Points: Proof by Induction5 分
Sums of Numbers8 分
Bernoulli's Inequality8 分
Coins Problem9 分
Cutting a Triangle8 分
Flawed Induction Proofs9 分
Alternating Sum9 分
3件の学習用教材
Two Cells of Opposite Colors: Hints10 分
Slides1 分
Slides10 分
5の練習問題
Largest Amount that Cannot Be Paid with 5- and 7-Coins10 分
Pay Any Large Amount with 5- and 7-Coins20 分
Number of Moves to Solve the Hanoi Towers Puzzle30 分
Two Cells of Opposite Colors: Feedback
Induction18 分
4
3時間で修了

Logic

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件のビデオ (合計53分), 2 readings, 9 quizzes
10件のビデオ
Counterexamples4 分
Basic Logic Constructs10 分
If-Then Generalization, Quantification8 分
Reductio ad Absurdum4 分
Balls in Boxes4 分
Numbers in Tables5 分
Pigeonhole Principle2 分
An (-1,0,1) Antimagic Square2 分
Handshakes3 分
2件の学習用教材
Slides10 分
Slides1 分
4の練習問題
Examples, Counterexamples and Logic14 分
Numbers in Boxes5 分
How to Pick Socks5 分
Pigeonhole Principle10 分
4.4
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Mathematical Thinking in Computer Science からの人気レビュー

by 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.

by 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.

講師

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Alexander S. Kulikov

Visiting Professor
Department of Computer Science and Engineering
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Michael Levin

Lecturer
Computer Science
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Vladimir Podolskii

Associate Professor
Computer Science Department

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