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.
Este curso forma parte de Programa Especializado - Introduction to Discrete Mathematics for Computer Science
Mathematical Thinking in Computer Science
ofrecido por
Universidad de California en San Diego
National Research University Higher School of Economics
Programa Especializado - Introduction to Discrete Mathematics for Computer ScienceUniversidad de California en San Diego
Acerca de este Curso
4.4
422 calificaciones
Habilidades que obtendrás
Mathematical InductionProof TheoryDiscrete MathematicsMathematical Logic
Programa - Qué aprenderás en este curso
Semana
13 horas para completar
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 videos (Total 43 minutos), 4 readings, 4 quizzes
10 videos
Proofs?3m
Proof by Example1m
Impossibility Proof2m
Impossibility Proof, II and Conclusion3m
One Example is Enough3m
Splitting an Octagon1m
Making Fun in Real Life: Tensegrities10m
Know Your Rights5m
Nobody Can Win All The Time: Nonexisting Examples8m
4 lecturas
Slides10m
Python10m
Slides1m
Acknowledgements1m
1 ejercicio de práctica
Tiles, dominos, black and white, even and odd6m
Semana
25 horas para completar
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 videos (Total 90 minutos), 6 readings, 12 quizzes
16 videos
Narrowing the Search6m
Multiplicative Magic Squares5m
More Puzzles9m
Integer Linear Combinations5m
Paths In a Graph4m
N Queens: Brute Force Search (Optional)10m
N Queens: Backtracking: Example (Optional)7m
N Queens: Backtracking: Code (Optional)7m
16 Diagonals (Optional)3m
Warm-up5m
Subset without x and 100-x4m
Rooks on a Chessboard2m
Knights on a Chessboard5m
Bishops on a Chessboard2m
Subset without x and 2x6m
6 lecturas
Slides1m
N Queens: Brute Force Solution Code (Optional)10m
N Queens: Backtracking Solution Code (Optional)10m
16 Diagonals: Code (Optional)10m
Slides (Optional)1m
Slides1m
3 ejercicios de práctica
Is there...20m
Number of Solutions for the 8 Queens Puzzle (Optional)20m
Maximum Number of Two-digit Integers2m
Semana
36 horas para completar
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 videos (Total 111 minutos), 3 readings, 8 quizzes
13 videos
Coin Problem4m
Hanoi Towers7m
Introduction, Lines and Triangles Problem10m
Lines and Triangles: Proof by Induction5m
Connecting Points12m
Odd Points: Proof by Induction5m
Sums of Numbers8m
Bernoulli's Inequality8m
Coins Problem9m
Cutting a Triangle8m
Flawed Induction Proofs9m
Alternating Sum9m
3 lecturas
Two Cells of Opposite Colors: Hints10m
Slides1m
Slides10m
5 ejercicios de práctica
Largest Amount that Cannot Be Paid with 5- and 7-Coins10m
Pay Any Large Amount with 5- and 7-Coins20m
Number of Moves to Solve the Hanoi Towers Puzzle30m
Two Cells of Opposite Colors: Feedback
Induction18m
Semana
43 horas para completar
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 videos (Total 53 minutos), 2 readings, 9 quizzes
10 videos
Examples6m
Counterexamples4m
Basic Logic Constructs10m
If-Then Generalization, Quantification8m
Reductio ad Absurdum4m
Balls in Boxes4m
Numbers in Tables5m
Pigeonhole Principle2m
An (-1,0,1) Antimagic Square2m
Handshakes3m
2 lecturas
Slides10m
Slides1m
4 ejercicios de práctica
Examples, Counterexamples and Logic14m
Numbers in Boxes5m
How to Pick Socks5m
Pigeonhole Principle10m
36%
comenzó una nueva carrera después de completar estos cursos
34%
consiguió un beneficio tangible en su carrera profesional gracias a este curso
Principales revisiones
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.
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.
Instructores
Alexander S. Kulikov
Visiting ProfessorDepartment of Computer Science and Engineering
Michael Levin
LecturerComputer Science
Vladimir Podolskii
Associate ProfessorComputer Science Department
Acerca de Universidad de California en 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.
Acerca de 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 communicamathematics, engineering, and more.
Learn more on www.hse.ru
Acerca del programa especializado 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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