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.

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Table of Contents (17 chapters)
Preface
Preface
Who this book is for
What this book covers
To get the most out of this book
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Conventions used
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1
Part I – Basic Concepts of Discrete Math
Part I – Basic Concepts of Discrete Math
Free Chapter
2
Chapter 1: Key Concepts, Notation, Set Theory, Relations, and Functions
Chapter 1: Key Concepts, Notation, Set Theory, Relations, and Functions
What is discrete mathematics?
Elementary set theory
Functions and relations
Summary
3
Chapter 2: Formal Logic and Constructing Mathematical Proofs
Chapter 2: Formal Logic and Constructing Mathematical Proofs
Formal Logic and Proofs by Truth Tables
Direct Mathematical Proofs
Proof by Contradiction
Proof by mathematical induction
Summary
4
Chapter 3: Computing with Base-n Numbers
Chapter 3: Computing with Base-n Numbers
Understanding base-n numbers
Converting between bases
Binary numbers and their applications
Hexadecimal numbers and their application
Summary
5
Chapter 4: Combinatorics Using SciPy
Chapter 4: Combinatorics Using SciPy
The fundamental counting rule
Counting permutations and combinations of objects
Applications to memory allocation
Efficacy of brute-force algorithms
Summary
6
Chapter 5: Elements of Discrete Probability
Chapter 5: Elements of Discrete Probability
The basics of discrete probability
Conditional probability and Bayes' theorem
Bayesian spam filtering
Random variables, means, and variance
Google PageRank I
Summary
7
Part II – Implementing Discrete Mathematics in Data and Computer Science
Part II – Implementing Discrete Mathematics in Data and Computer Science
8
Chapter 6: Computational Algorithms in Linear Algebra
Chapter 6: Computational Algorithms in Linear Algebra
Understanding linear systems of equations
Matrices and matrix representations of linear systems
Solving small linear systems with Gaussian elimination
Solving large linear systems with NumPy
Summary
9
Chapter 7: Computational Requirements for Algorithms
Chapter 7: Computational Requirements for Algorithms
Computational complexity of algorithms
Understanding Big-O Notation
Complexity of algorithms with fundamental control structures
Complexity of common search algorithms
Common classes of computational complexity
Summary
References
10
Chapter 8: Storage and Feature Extraction of Graphs, Trees, and Networks
Chapter 8: Storage and Feature Extraction of Graphs, Trees, and Networks
Understanding graphs, trees, and networks
Using graphs, trees, and networks
Storage of graphs and networks
Feature extraction of graphs
Summary
11
Chapter 9: Searching Data Structures and Finding Shortest Paths
Chapter 9: Searching Data Structures and Finding Shortest Paths
Searching Graph and Tree data structures
Depth-first search (DFS)
The shortest path problem and variations of the problem
Finding Shortest Paths with Brute Force
Dijkstra's Algorithm for Finding Shortest Paths
Python Implementation of Dijkstra's Algorithm
Summary
12
Part III – Real-World Applications of Discrete Mathematics
Part III – Real-World Applications of Discrete Mathematics
13
Chapter 10: Regression Analysis with NumPy and Scikit-Learn
Chapter 10: Regression Analysis with NumPy and Scikit-Learn
Dataset
Best-fit lines and the least-squares method
Least-squares lines with NumPy
Least-squares curves with NumPy and SciPy
Least-squares surfaces with NumPy and SciPy
Summary
14
Chapter 11: Web Searches with PageRank
Chapter 11: Web Searches with PageRank
The Development of Search Engines over time
Google PageRank II
Implementing the PageRank algorithm in Python
Applying the Algorithm to Real Data
Summary
15
Chapter 12: Principal Component Analysis with Scikit-Learn
Chapter 12: Principal Component Analysis with Scikit-Learn
Understanding eigenvalues, eigenvectors, and orthogonal bases
The principal component analysis approach to dimensionality reduction
The scikit-learn implementation of PCA
An application to real-world data
Summary
16
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What this book covers

Part I – Basic Concepts of Discrete Math

Chapter 1, Key Concepts, Notation, Set Theory, Relations, and Functions, is an introduction to the basic vocabulary, concepts, and notation of discrete mathematics.

Chapter 2, Formal Logic and Constructing Mathematical Proofs, covers formal logic and binary and explains how to prove mathematical results.

Chapter 3, Computing with Base-n Numbers, discusses arithmetic in different numbering systems, including hexadecimal and binary.

Chapter 4, Combinatorics Using SciPy, explains how to count the elements in certain types of discrete structures.

Chapter 5, Elements of Discrete Probability, covers measuring chance and the basics of Google's PageRank algorithm.

Part II – Implementing Discrete Mathematics in Data and Computer Science

Chapter 6, Computational Algorithms in Linear Algebra, explains how to solve algebra problems with Python using NumPy.

Chapter 7, Computational Requirements for Algorithms, gives you the tools to determine how long algorithms take to run and how much space they require.

Chapter 8, Storage and Feature Extraction of Graphs, Trees, and Networks, covers storing graph structures and finding information about them with code.

Chapter 9, Searching Data Structures and Finding Shortest Paths, explains how to traverse graphs and figure out efficient paths between vertices.

Part III – Real-World Applications of Discrete Mathematics

Chapter 10, Regression Analysis with NumPy, is a discussion on the prediction of variables in datasets containing multiple variables.

Chapter 11, Web Searches with PageRank, shows you how to rank the results of web searches to find the most relevant web pages.

Chapter 12, Principal Component Analysis with Scikit-Learn, explains how to reduce the dimensionality of high-dimensional datasets to save space and speed up machine learning.

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