Practical Discrete Mathematics

Ryan T. White, Archana Tikayat Ray

• Publisher: Packt Publishing
• Year: 2021
• ISBN:
  9781838983505
• Edition: 1
• Topic: Mathematics
• Subtopic: Discrete Mathematics
• Index: Mathematics

A practical guide simplifying discrete math for curious minds and demonstrating its application in solving problems related to software development, computer algorithms, and data scienceKey Features• Apply the math of countable objects to practical problems in computer science• Explore modern Python libraries such as scikit-learn, NumPy, and SciPy for performing mathematics• Learn complex statistical and mathematical concepts with the help of hands-on examples and expert guidanceBook DescriptionDiscrete mathematics deals with studying countable, distinct elements, and its principles are widely used in building algorithms for computer science and data science. The knowledge of discrete math concepts will help you understand the algorithms, binary, and general mathematics that sit at the core of data-driven tasks. Practical Discrete Mathematics is a comprehensive introduction for those who are new to the mathematics of countable objects. This book will help you get up to speed with using discrete math principles to take your computer science skills to a more advanced level. As you learn the language of discrete mathematics, you'll also cover methods crucial to studying and describing computer science and machine learning objects and algorithms. The chapters that follow will guide you through how memory and CPUs work. In addition to this, you'll understand how to analyze data for useful patterns, before finally exploring how to apply math concepts in network routing, web searching, and data science. By the end of this book, you'll have a deeper understanding of discrete math and its applications in computer science, and be ready to work on real-world algorithm development and machine learning.What you will learn• Understand the terminology and methods in discrete math and their usage in algorithms and data problems• Use Boolean algebra in formal logic and elementary control structures• Implement combinatorics to measure computational complexity and manage memory allocation• Use random variables, calculate descriptive statistics, and find average-case computational complexity• Solve graph problems involved in routing, pathfinding, and graph searches, such as depth-first search• Perform ML tasks such as data visualization, regression, and dimensionality reductionWho this book is forThis book is for computer scientists looking to expand their knowledge of discrete math, the core topic of their field. University students looking to get hands-on with computer science, mathematics, statistics, engineering, or related disciplines will also find this book useful. Basic Python programming skills and knowledge of elementary real-number algebra are required to get started with this book.

Part I – Basic Concepts of Discrete Math

Here you will learn the critical vocabulary, notations, and methods of discrete mathematics, including set theory, functions and relations, logic and proofs, arithmetic, counting, and basic probability as applied to computer science.

This part comprises the following chapters:

• Chapter 1, Key Concepts, Notation, Set Theory, Relations, and Functions
• Chapter 2, Formal Logic and Constructing Mathematical Proofs
• Chapter 3, Computing with Base-n Numbers
• Chapter 4, Combinatorics Using SciPy
• Chapter 5, Elements of Discrete Probability

Chapter 1: Key Concepts, Notation, Set Theory, Relations, and Functions

This chapter is a general introduction to the main ideas of discrete mathematics. Alongside this, we will go through key terms and concepts in the field. After that, we will cover set theory, the essential notation and notions for referring to collections of mathematical objects and combining or selecting them. We will also think about mapping mathematical objects to one another with functions and relations and visualizing them with graphs.

In this chapter, we will cover the following topics:

• What is discrete mathematics?
• Elementary set theory
• Functions and relations

By the end of the chapter, you should be able to speak in the language of discrete mathematics and understand notation common to the entire field.

Important Note

Please navigate to the graphic bundle link to refer to the color images for this chapter.

What is discrete mathematics?

Discrete mathematics is the study of countable, distinct, or separate mathematical structures. A good example is a pixel. From phones to computer monitors to televisions, modern screens are made up of millions of tiny dots called pixels lined up in grids. Each pixel lights up with a specified color on command from a device, but only a finite number of colors can be displayed in each pixel.

The millions of colored dots taken together form intricate patterns and give our eyes the impression of shapes with smooth curves, as in the boundary of the following circle:

Figure 1.1 – The boundary of a circle

But if you zoom in and look closely enough, the true "curves" are revealed to be jagged boundaries between differently colored regions of pixels, possibly with some intermediate colors, as shown in the following diagram:

Figure 1.2 – A zoomed-in view of the circle

Some other examples of objects studied in discrete mathematics are logical statements, integers, bits and bytes, graphs, trees, and networks. Like pixels, these too can form intricate patterns that we will try to discover and exploit for various purposes related to computer and data scienc...