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初級

完成時間大約為25 小時

建議:6 weeks, 2–5 hours/week...

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Mathematical InductionProof TheoryDiscrete MathematicsMathematical Logic

100% 在線

立即開始,按照自己的計劃學習。

可靈活調整截止日期

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初級

完成時間大約為25 小時

建議:6 weeks, 2–5 hours/week...

英語(English)

字幕:英語(English)

教學大綱 - 您將從這門課程中學到什麼

1
完成時間為 3 小時

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.

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10 個視頻 (總計 43 分鐘), 4 個閱讀材料, 4 個測驗
10 個視頻
Proofs?3分鐘
Proof by Example1分鐘
Impossibility Proof2分鐘
Impossibility Proof, II and Conclusion3分鐘
One Example is Enough3分鐘
Splitting an Octagon1分鐘
Making Fun in Real Life: Tensegrities10分鐘
Know Your Rights5分鐘
Nobody Can Win All The Time: Nonexisting Examples8分鐘
4 個閱讀材料
Slides10分鐘
Python10分鐘
Slides1分鐘
Acknowledgements1分鐘
1 個練習
Tiles, dominos, black and white, even and odd6分鐘
2
完成時間為 5 小時

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.

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16 個視頻 (總計 90 分鐘), 6 個閱讀材料, 12 個測驗
16 個視頻
Narrowing the Search6分鐘
Multiplicative Magic Squares5分鐘
More Puzzles9分鐘
Integer Linear Combinations5分鐘
Paths In a Graph4分鐘
N Queens: Brute Force Search (Optional)10分鐘
N Queens: Backtracking: Example (Optional)7分鐘
N Queens: Backtracking: Code (Optional)7分鐘
16 Diagonals (Optional)3分鐘
Warm-up5分鐘
Subset without x and 100-x4分鐘
Rooks on a Chessboard2分鐘
Knights on a Chessboard5分鐘
Bishops on a Chessboard2分鐘
Subset without x and 2x6分鐘
6 個閱讀材料
Slides1分鐘
N Queens: Brute Force Solution Code (Optional)10分鐘
N Queens: Backtracking Solution Code (Optional)10分鐘
16 Diagonals: Code (Optional)10分鐘
Slides (Optional)1分鐘
Slides1分鐘
3 個練習
Is there...20分鐘
Number of Solutions for the 8 Queens Puzzle (Optional)20分鐘
Maximum Number of Two-digit Integers2分鐘
3
完成時間為 6 小時

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 個視頻 (總計 111 分鐘), 3 個閱讀材料, 8 個測驗
13 個視頻
Recursion9分鐘
Coin Problem4分鐘
Hanoi Towers7分鐘
Introduction, Lines and Triangles Problem10分鐘
Lines and Triangles: Proof by Induction5分鐘
Connecting Points12分鐘
Odd Points: Proof by Induction5分鐘
Sums of Numbers8分鐘
Bernoulli's Inequality8分鐘
Coins Problem9分鐘
Cutting a Triangle8分鐘
Flawed Induction Proofs9分鐘
Alternating Sum9分鐘
3 個閱讀材料
Two Cells of Opposite Colors: Hints10分鐘
Slides1分鐘
Slides10分鐘
5 個練習
Largest Amount that Cannot Be Paid with 5- and 7-Coins10分鐘
Pay Any Large Amount with 5- and 7-Coins20分鐘
Number of Moves to Solve the Hanoi Towers Puzzle30分鐘
Two Cells of Opposite Colors: Feedback
Induction18分鐘
4
完成時間為 3 小時

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 個視頻 (總計 53 分鐘), 2 個閱讀材料, 9 個測驗
10 個視頻
Examples6分鐘
Counterexamples4分鐘
Basic Logic Constructs10分鐘
If-Then Generalization, Quantification8分鐘
Reductio ad Absurdum4分鐘
Balls in Boxes4分鐘
Numbers in Tables5分鐘
Pigeonhole Principle2分鐘
An (-1,0,1) Antimagic Square2分鐘
Handshakes3分鐘
2 個閱讀材料
Slides10分鐘
Slides1分鐘
4 個練習
Examples, Counterexamples and Logic14分鐘
Numbers in Boxes5分鐘
How to Pick Socks5分鐘
Pigeonhole Principle10分鐘
4.4
112 個審閱Chevron Right

41%

完成這些課程後已開始新的職業生涯

38%

通過此課程獲得實實在在的工作福利

來自Mathematical Thinking in Computer Science的熱門評論

創建者 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.

創建者 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

關於 加州大学圣地亚哥分校

UC San Diego is an academic powerhouse and economic engine, recognized as one of the top 10 public universities by U.S. News and World Report. Innovation is central to who we are and what we do. Here, students learn that knowledge isn't just acquired in the classroom—life is their laboratory....

關於 国立高等经济大学

National Research University - Higher School of Economics (HSE) is one of the top research universities in Russia. Established in 1992 to promote new research and teaching in economics and related disciplines, it now offers programs at all levels of university education across an extraordinary range of fields of study including business, sociology, cultural studies, philosophy, political science, international relations, law, Asian studies, media and communicamathematics, engineering, and more. Learn more on www.hse.ru...

關於 Introduction to Discrete Mathematics for Computer Science 專項課程

Discrete Math is needed to see mathematical structures in the object you work with, and understand their properties. This ability is important for software engineers, data scientists, security and financial analysts (it is not a coincidence that math puzzles are often used for interviews). We cover the basic notions and results (combinatorics, graphs, probability, number theory) that are universally needed. To deliver techniques and ideas in discrete mathematics to the learner we extensively use interactive puzzles specially created for this specialization. To bring the learners experience closer to IT-applications we incorporate programming examples, problems and projects in our courses....
Introduction to Discrete Mathematics for Computer Science

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