课程信息
4.4
337 个评分
77 个审阅
100% 在线

100% 在线

立即开始,按照自己的计划学习。
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初级

初级

完成时间(小时)

完成时间大约为20 小时

建议:6 weeks, 2–5 hours/week...
可选语言

英语(English)

字幕:英语(English)

您将获得的技能

Mathematical InductionProof TheoryDiscrete MathematicsMathematical Logic
100% 在线

100% 在线

立即开始,按照自己的计划学习。
可灵活调整截止日期

可灵活调整截止日期

根据您的日程表重置截止日期。
初级

初级

完成时间(小时)

完成时间大约为20 小时

建议:6 weeks, 2–5 hours/week...
可选语言

英语(English)

字幕:英语(English)

教学大纲 - 您将从这门课程中学到什么

1
完成时间(小时)
完成时间为 3 小时

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....
Reading
10 个视频 (总计 43 分钟), 4 个阅读材料, 4 个测验
Video10 个视频
Proofs?3分钟
Proof by Example1分钟
Impossibility Proof2分钟
Impossibility Proof, II and Conclusion3分钟
One Example is Enough3分钟
Splitting an Octagon1分钟
Making Fun in Real Life: Tensegrities10分钟
Know Your Rights5分钟
Nobody Can Win All The Time: Nonexisting Examples8分钟
Reading4 个阅读材料
Slides10分钟
Python10分钟
Slides1分钟
Acknowledgements1分钟
Quiz1 个练习
Tiles, dominos, black and white, even and odd6分钟
2
完成时间(小时)
完成时间为 5 小时

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....
Reading
16 个视频 (总计 90 分钟), 5 个阅读材料, 12 个测验
Video16 个视频
Narrowing the Search6分钟
Multiplicative Magic Squares5分钟
More Puzzles9分钟
Integer Linear Combinations5分钟
Paths In a Graph4分钟
N Queens: Brute Force Search (Optional)10分钟
N Queens: Backtracking: Example (Optional)7分钟
N Queens: Backtracking: Code (Optional)7分钟
16 Diagonals (Optional)3分钟
Warm-up5分钟
Subset without x and 100-x4分钟
Rooks on a Chessboard2分钟
Knights on a Chessboard5分钟
Bishops on a Chessboard2分钟
Subset without x and 2x6分钟
Reading5 个阅读材料
Slides1分钟
N Queens: Brute Force Solution Code (Optional)10分钟
N Queens: Backtracking Solution Code (Optional)10分钟
Slides (Optional)1分钟
Slides1分钟
Quiz3 个练习
Is there...20分钟
Number of Solutions for the 8 Queens Puzzle (Optional)20分钟
Maximum Number of Two-digit Integers2分钟
3
完成时间(小时)
完成时间为 5 小时

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....
Reading
13 个视频 (总计 111 分钟), 2 个阅读材料, 7 个测验
Video13 个视频
Recursion9分钟
Coin Problem4分钟
Hanoi Towers7分钟
Introduction, Lines and Triangles Problem10分钟
Lines and Triangles: Proof by Induction5分钟
Connecting Points12分钟
Odd Points: Proof by Induction5分钟
Sums of Numbers8分钟
Bernoulli's Inequality8分钟
Coins Problem9分钟
Cutting a Triangle8分钟
Flawed Induction Proofs9分钟
Alternating Sum9分钟
Reading2 个阅读材料
Slides1分钟
Slides10分钟
Quiz4 个练习
Largest Amount that Cannot Be Paid with 5- and 7-Coins10分钟
Pay Any Large Amount with 5- and 7-Coins20分钟
Number of Moves to Solve the Hanoi Towers Puzzle30分钟
Induction18分钟
4
完成时间(小时)
完成时间为 3 小时

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....
Reading
10 个视频 (总计 53 分钟), 2 个阅读材料, 9 个测验
Video10 个视频
Examples6分钟
Counterexamples4分钟
Basic Logic Constructs10分钟
If-Then Generalization, Quantification8分钟
Reductio ad Absurdum4分钟
Balls in Boxes4分钟
Numbers in Tables5分钟
Pigeonhole Principle2分钟
An (-1,0,1) Antimagic Square2分钟
Handshakes3分钟
Reading2 个阅读材料
Slides10分钟
Slides1分钟
Quiz4 个练习
Examples, Counterexamples and Logic14分钟
Numbers in Boxes5分钟
How to Pick Socks5分钟
Pigeonhole Principle10分钟
4.4
77 个审阅Chevron Right
职业方向

60%

完成这些课程后已开始新的职业生涯
工作福利

60%

通过此课程获得实实在在的工作福利

热门审阅

创建者 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.

创建者 KLNov 29th 2018

The course is excellent and most stuff is being taught in a nicely presented way. The main disappointment is 15-puzzle, because it's too difficult to understand without proper material.

讲师

Avatar

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

关于 University of California San Diego

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....

关于 National Research University Higher School of Economics

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 communications, IT, mathematics, engineering, and more. Learn more on www.hse.ru...

关于 Introduction to Discrete Mathematics for Computer Science 专项课程

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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