Book Image

Practical Discrete Mathematics

By : Ryan T. White, Archana Tikayat Ray
Book Image

Practical Discrete Mathematics

By: Ryan T. White, Archana Tikayat Ray

Overview of this book

Discrete 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.
Table of Contents (17 chapters)
1
Part I – Basic Concepts of Discrete Math
7
Part II – Implementing Discrete Mathematics in Data and Computer Science
12
Part III – Real-World Applications of Discrete Mathematics

Proof by Contradiction

In this section, we will learn about using contradiction for mathematical proofs. Proof by contradiction is a method of proof where you first assume the claim you wish to prove is false, and then prove through a series of logical deductions that this assumption results in a contradictory claim. If this happens, and we have made no errors, this assumption that the claim was false must have been incorrect. Thus, the claim must be true.

While this idea may make sense abstractly and we see the proof method is confirmed by formal logic, the authors believe the method is best demonstrated by examples if you hope to build some intuitive understanding of the approach, learn when it is likely to be effective, and construct your own mathematical proofs.

First, let's review some ideas we all probably learned in primary school. Recall a real number x is called rational if it can be written as a ratio:

Here, a and b ≠ 0 are relatively...