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?

This course is part of the Introduction to Discrete Mathematics for Computer Science Specialization

Offered By

## About this Course

### Learner Career Outcomes

## 35%

## 32%

### Skills you will gain

### Learner Career Outcomes

## 35%

## 32%

### Offered by

#### 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.

#### 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.

## Syllabus - What you will learn from this course

**3 hours to complete**

## 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.

**3 hours to complete**

**10 videos**

**5 readings**

**1 practice exercise**

**6 hours to complete**

## 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.

**6 hours to complete**

**16 videos**

**6 readings**

**3 practice exercises**

**7 hours to complete**

## 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 discrete mathematics and computer science. In particular, you will see them frequently in algorithms — for analysing correctness and running time of algorithms as well as for implementing efficient solutions. For some computational problems (e.g., exploring networks), recursive solutions are the most natural ones. The main idea of recursion and induction is to decompose a given problem into smaller problems of the same type. Being able to see such decompositions is an important skill both in mathematics and in programming. We'll hone this skill by solving various problems together.

**7 hours to complete**

**13 videos**

**3 readings**

**5 practice exercises**

**4 hours to complete**

## 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.

**4 hours to complete**

**10 videos**

**2 readings**

**4 practice exercises**

## Reviews

### TOP REVIEWS FROM MATHEMATICAL THINKING IN COMPUTER SCIENCE

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 loved this course! So many interesting things to think about, thoughtfully explained by brilliant instructors. The puzzles really get you thinking. Such genius to put them before the lectures!

I applaud the instructors for their efforts in explaining the concepts as they could be abstract and hard to explain in words! More examples to illustrate the concepts will be even more helpful!

I have come to know how mathematical proof is fun thing to do, this course transformed me, i highly recommend it to every one. I would like to thank every one involved in providing this course.

## About the Introduction to Discrete Mathematics for Computer Science Specialization

Discrete Mathematics is the language of Computer Science. One needs to be fluent in it to work in many fields including data science, machine learning, and software engineering (it is not a coincidence that math puzzles are often used for interviews). We introduce you to this language through a fun try-this-before-we-explain-everything approach: first you solve many interactive puzzles that are carefully designed specifically for this online specialization, and then we explain how to solve the puzzles, and introduce important ideas along the way. We believe that this way, you will get a deeper understanding and will better appreciate the beauty of the underlying ideas (not to mention the self confidence that you gain if you invent these ideas on your own!). To bring your experience closer to IT-applications, we incorporate programming examples, problems, and projects in the specialization.

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