Mathematical Thinking in Computer Science

Go to Course: https://www.coursera.org/learn/what-is-a-proof

Introduction

## Course Review: Mathematical Thinking in Computer Science ### Overview In the digital age, mathematical reasoning is a cornerstone of computer science, underpinning everything from algorithms to data science, bioinformatics, and machine learning. The **"Mathematical Thinking in Computer Science"** course on Coursera provides an in-depth exploration of the essential tools that empower computer scientists to tackle complex problems through effective reasoning and structured thinking. Designed for learners who wish to enhance their problem-solving skills, this course focuses on discrete mathematics concepts such as induction, recursion, logic, invariants, and optimality. Each concept is tied to real-world programming questions, making the material relevant and applicable. ### Syllabus Highlights 1. **Making Convincing Arguments** The course kicks off with a critical examination of what makes an argument persuasive. It explores the art of establishing unassailable proofs and demonstrates the joy of mathematical reasoning. You'll engage in analyzing and constructing proofs, gaining an appreciation for how even simple observations can illuminate complex questions. 2. **How to Find an Example?** This module teaches you to ascertain the existence of objects based on given requirements. Given the vastness of potential solutions, students learn techniques to narrow down search spaces effectively. The module is interactive, requiring participants to solve puzzles and develop programs to construct examples, enhancing practical understanding. 3. **Recursion and Induction** Here, the course delves into two powerful strategies: recursion and induction. These concepts are pivotal for defining objects and proving the correctness of algorithms. By breaking down problems into smaller, manageable pieces, students practice these techniques, honing their proficiency in both mathematical reasoning and programming. 4. **Logic** A fundamental component of mathematical thought, this section introduces the basics of mathematical logic. Understanding logical propositions and arguments is essential for writing clean, efficient code and making sound decisions in programming contexts. 5. **Invariants** The principle that certain properties remain unchanged during processes is explored in-depth in this module. Invariants are crucial for analyzing and understanding algorithms, and you will develop the skill to identify and utilize them effectively. 6. **Solving a 15-Puzzle** The course culminates in a practical application: solving a 15-puzzle using mathematical concepts. Students will explore the significance of permutations in algebra and how this relates directly to puzzle solvability. This hands-on project not only solidifies learning but also involves coding solutions based on the mathematical principles discussed throughout the course. ### Recommendations This course is an exceptional fit for anyone looking to deepen their understanding of mathematical concepts in computer science. Here are a few recommendations: - **For Beginners**: If you’re new to computer science or looking to solidify your foundational knowledge, this course is accessible and structured in a way that builds your confidence gradually. - **For Advanced Learners**: Those with a firm grasp of basic programming and mathematics will find the course enriching, providing advanced techniques and insights that can enhance their existing skill set. - **Practical Application**: If you’re a practitioner or student in fields like data science, algorithms, or machine learning, the real-world applications discussed will resonate and enhance your professional abilities. - **Self-Motivated Learners**: The interactive elements and hands-on projects ensure that you will not only learn but also apply your knowledge, making this an engaging learning experience. ### Conclusion "Mathematical Thinking in Computer Science" is a comprehensive course that effectively marries the theoretical aspects of mathematics with practical programming applications. By the end of the course, you will not only master the essential tools of mathematical thinking but also apply them with confidence in your computer science endeavors. Whether you are a budding programmer or an experienced developer, this course promises to elevate your problem-solving skills to new heights. Invest in your mathematical and programming education today and join the countless others who have benefitted from the insights offered in this essential course on Coursera!

Syllabus

Making Convincing Arguments

Why are some arguments convincing and some others are not? What makes an argument convincing? How can you establish your argument in such a way that there is no room for doubt left? How can mathematical thinking help with this? In this section, we start digging into these questions. Our goal is to learn by examples how to understand proofs, how to discover them on your own, how to explain them, and — last but not least — how to enjoy them: we will see how a small remark or a simple observation can turn a seemingly non-trivial question into an obvious one.

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.

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.

Mathematical logic plays a crucial and indispensable role in creating convincing arguments. We use the rules and language of mathematical logic while writing code, while reasoning and making decisions, and while using computer programs. This week we’ll learn the basics of mathematical logic, and we'll practice tricky and seemingly counterintuitive, but yet logical aspects of mathematical logic. This will help us to write readable and precise code, and to formulate our thoughts rigorously and concisely.

"There are things that never change". Apart from being just a philosophical statement, this phrase turns out to be an important idea in discrete mathematics and computer science. A property that is preserved during a process is called an invariant. Invariants are used heavily in analyzing the behavior of algorithms, programs, and other processes. Being able to find the right invariant is an important skill that we will develop together in this module.

In this module, we consider a well known 15-puzzle where one needs to restore order among 15 square pieces in a square box. It turns out that the behavior of this puzzle is determined by beautiful mathematics: it is solvable if and only if the corresponding permutation is even. To understand what it means and why it is true, we will learn the basic properties of even and odd permutations — an important notion in algebra and discrete mathematics. Together, we will implement a number of simple methods for working with permutations. You will then use them as building blocks to implement a program that solves any configuration of this game in blink of an eye!