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?
In the course, we use a try-this-before-we-explain-everything approach: you will be solving many interactive (and mobile friendly) puzzles that were carefully designed to allow you to invent many of the important ideas and concepts yourself.
Prerequisites:
1. We assume only basic math (e.g., we expect you to know what is a square or how to add fractions), common sense and curiosity.
2. Basic programming knowledge is necessary as some quizzes require programming in Python.
Ce cours fait partie de la Spécialisation Introduction to Discrete Mathematics for Computer Science
Mathematical Thinking in Computer Science
proposé par
Université de Californie à San Diego
Université nationale de recherche, École des hautes études en sciences économiques
Spécialisation Introduction to Discrete Mathematics for Computer ScienceUniversité de Californie à San Diego
À propos de ce cours
4.4
455 notes
Compétences que vous acquerrez
Mathematical InductionProof TheoryDiscrete MathematicsMathematical Logic
Programme du cours : ce que vous apprendrez dans ce cours
Semaine
13 heures pour terminer
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 vidéos (Total 43 min), 4 lectures, 4 quiz
10 vidéos
Promo Video1 min
Proofs?3 min
Proof by Example1 min
Impossibility Proof2 min
Impossibility Proof, II and Conclusion3 min
One Example is Enough3 min
Splitting an Octagon1 min
Making Fun in Real Life: Tensegrities10 min
Know Your Rights5 min
Nobody Can Win All The Time: Nonexisting Examples8 min
4 lectures
Slides10 min
Python10 min
Slides1 min
Acknowledgements1 min
1 exercice pour s'entraîner
Tiles, dominos, black and white, even and odd6 min
Semaine
25 heures pour terminer
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 vidéos (Total 90 min), 6 lectures, 12 quiz
16 vidéos
Magic Squares3 min
Narrowing the Search6 min
Multiplicative Magic Squares5 min
More Puzzles9 min
Integer Linear Combinations5 min
Paths In a Graph4 min
N Queens: Brute Force Search (Optional)10 min
N Queens: Backtracking: Example (Optional)7 min
N Queens: Backtracking: Code (Optional)7 min
16 Diagonals (Optional)3 min
Warm-up5 min
Subset without x and 100-x4 min
Rooks on a Chessboard2 min
Knights on a Chessboard5 min
Bishops on a Chessboard2 min
Subset without x and 2x6 min
6 lectures
Slides1 min
N Queens: Brute Force Solution Code (Optional)10 min
N Queens: Backtracking Solution Code (Optional)10 min
16 Diagonals: Code (Optional)10 min
Slides (Optional)1 min
Slides1 min
3 exercices pour s'entraîner
Is there...20 min
Number of Solutions for the 8 Queens Puzzle (Optional)20 min
Maximum Number of Two-digit Integers2 min
Semaine
36 heures pour terminer
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 vidéos (Total 111 min), 3 lectures, 8 quiz
13 vidéos
Recursion9 min
Coin Problem4 min
Hanoi Towers7 min
Introduction, Lines and Triangles Problem10 min
Lines and Triangles: Proof by Induction5 min
Connecting Points12 min
Odd Points: Proof by Induction5 min
Sums of Numbers8 min
Bernoulli's Inequality8 min
Coins Problem9 min
Cutting a Triangle8 min
Flawed Induction Proofs9 min
Alternating Sum9 min
3 lectures
Two Cells of Opposite Colors: Hints10 min
Slides1 min
Slides10 min
5 exercices pour s'entraîner
Largest Amount that Cannot Be Paid with 5- and 7-Coins10 min
Pay Any Large Amount with 5- and 7-Coins20 min
Number of Moves to Solve the Hanoi Towers Puzzle30 min
Two Cells of Opposite Colors: Feedback
Induction18 min
Semaine
43 heures pour terminer
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 vidéos (Total 53 min), 2 lectures, 9 quiz
10 vidéos
Examples6 min
Counterexamples4 min
Basic Logic Constructs10 min
If-Then Generalization, Quantification8 min
Reductio ad Absurdum4 min
Balls in Boxes4 min
Numbers in Tables5 min
Pigeonhole Principle2 min
An (-1,0,1) Antimagic Square2 min
Handshakes3 min
2 lectures
Slides10 min
Slides1 min
4 exercices pour s'entraîner
Examples, Counterexamples and Logic14 min
Numbers in Boxes5 min
How to Pick Socks5 min
Pigeonhole Principle10 min
4.4
96 avis36%
a commencé une nouvelle carrière après avoir terminé ces cours
34%
a bénéficié d'un avantage concret dans sa carrière grâce à ce cours
Meilleurs avis
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.
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.
Enseignants
Alexander S. Kulikov
Visiting ProfessorDepartment of Computer Science and Engineering
Michael Levin
LecturerComputer Science
Vladimir Podolskii
Associate ProfessorComputer Science Department
À propos de Université de Californie à 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.
À propos de Université nationale de recherche, École des hautes études en sciences économiques
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
À propos de la Spécialisation Introduction to Discrete Mathematics for Computer Science
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.
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