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

# Mathematical Thinking in Computer Science

Offered By

## About this Course

### Learner Career Outcomes

## 41%

## 37%

### Skills you will gain

### Learner Career Outcomes

## 41%

## 37%

#### 100% online

#### Course 1 of 5 in the

#### Flexible deadlines

#### Beginner Level

#### Approx. 25 hours to complete

#### English

## Syllabus - What you will learn from this course

**2 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.

**2 hours to complete**

**10 videos**

**4 readings**

**1 practice exercise**

**5 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.

**5 hours to complete**

**16 videos**

**6 readings**

**3 practice exercises**

**6 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 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.

**6 hours to complete**

**13 videos**

**3 readings**

**5 practice exercises**

**3 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.

**3 hours to complete**

**10 videos**

**2 readings**

**4 practice exercises**

### Reviews

#### 4.4

##### 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 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 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!

Contents are very good for starting.....\n\nBut the Teachers way of explaining is not up to the mark. I need to search in youTube or google for understanding any topic even watching the videos.

### About University of California San Diego

### About National Research University Higher School of Economics

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

## Frequently Asked Questions

When will I have access to the lectures and assignments?

Once you enroll for a Certificate, you’ll have access to all videos, quizzes, and programming assignments (if applicable). Peer review assignments can only be submitted and reviewed once your session has begun. If you choose to explore the course without purchasing, you may not be able to access certain assignments.

What will I get if I subscribe to this Specialization?

When you enroll in the course, you get access to all of the courses in the Specialization, and you earn a certificate when you complete the work. Your electronic Certificate will be added to your Accomplishments page - from there, you can print your Certificate or add it to your LinkedIn profile. If you only want to read and view the course content, you can audit the course for free.

What is the refund policy?

Is financial aid available?

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