Digital library
Textbooks, references and open learning resources.
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An Introduction to Proof via Inquiry-Based Learning
Dana Ernst · Northern Arizona University
This book is intended to be used for a one-semester/quarter introduction to proof course (sometimes referred to as a transition to proof course). The purpose of this book is to introduce the reader to the process of constructing and writing formal and rigorous mathematical proofs. The intended audience is mathematics majors and minors. However, this book is also appropriate for anyone curious about mathematics and writing proofs.
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Transition to Higher Mathematics
Bob Dumas and John E. McCarthy · University of Washington and Washington University in St. Louis
In the contents: 1.5: Images and Inverses
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Gentle Introduction to the Art of Mathematics
Joseph Fields · Southern Connecticut State University
In the contents: 01: Introduction and Notation · 1.06: Rational And Irrational Numbers · 02: Logic and Quantifiers
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Friendly Introduction to Mathematical Logic
Christopher Leary & Lars Kristiansen · SUNY Geneseo & University of Oslo
In the contents: 01: Structures and Languages · 1.04: Terms and Formulas · 1.09: Substitutions and Substitutability
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MCC: MTH 220 - Discrete Math
Judy Dean · Monroe Community College
In the contents: 1.3: How to Read and Write Mathematics · 1.5: Introduction to Sets and Real Numbers · 2: Logic
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Proofs and Concepts - The Fundamentals of Abstract Mathematics
Dave Witte Morris & Joy Morris · University of Lethbridge
In the contents: 01: Propositional Logic · 1.01: Assertions deductions and validity · 1.02: Logic Puzzles
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Euclid's Elements Redux
Daniel Callahan · Sample · 2020
"Euclid's 'Elements' Redux" is an open textbook on mathematical logic and geometry based on Euclid's "Elements" for use in grades 7-12 and in undergraduate college courses on proof writing. Many problem solvers throughout history wrestled with Euclid as part of their early education including Copernicus, Kepler, Galileo, Sir Isaac Newton, Ada Lovelace, Abraham Lincoln, Bertrand Russell, and Albert Einstein. This edition is part of an effort to ensure that tomorrow's great thinkers will have that same privilege.
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Proofs and Concepts: The Fundamentals of Abstract Mathematics
Dave Morris; Joy Morris · Dave Witte Morris, Joy Morris · 2013
This free undergraduate textbook provides an introduction to proofs, logic, sets, functions, and other fundamental topics of abstract mathematics. It is designed to be the textbook for a bridge course that introduces undergraduates to abstract mathematics, but it is also suitable for independent study by undergraduates (or mathematically mature high-school students), or for use as a very inexpensive supplement to undergraduate courses in any field of abstract mathematics.
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Discrete Mathematics: An Open Introduction
Oscar Levin · Oscar Levin · 2024
Discrete Mathematics: An Open Introduction is a free, open source textbook appropriate for a first or second year undergraduate course for math and computer science majors. The book is especially well-suited for courses that incorporate inquiry-based learning. Since Spring 2013, the book has been used as the primary textbook or a supplemental resource at more than 200 colleges and universities around the world (see the partial adoptions list). After many years of development, I am pleased to announce that the 4th edition of Discrete Mathematics: an Open Introduction is now available, here and on Runestone Academy. The new edition brings many improvements and a new organization of content. In particular, the book now starts with logic and proofs, then practices those proofs with graph theory. The second half of the book contains material on counting (with a new "application to probability" section) and sequences. Over the last few years, I have found that students have more success with this arrangement. There is also a stronger emphasis on discrete structures, which should make the book more useful for students in computer science, while still focusing on understanding mathematical concepts essential for math majors and future math teachers.
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Introduction to Mathematical Analysis I
Beatriz Lafferriere; Gerardo Lafferriere; Mau Nam Nguyen · Portland State University Library · 2016
Our goal with this textbook is to provide students with a strong foundation in mathematical analysis. Such a foundation is crucial for future study of deeper topics of analysis. Students should be familiar with most of the concepts presented here after completing the calculus sequence. However, these concepts will be reinforced through rigorous proofs. The lecture notes contain topics of real analysis usually covered in a 10-week course: the completeness axiom, sequences and convergence, continuity, and differentiation. The lecture notes also contain many well-selected exercises of various levels. Although these topics are written in a more abstract way compared with those available in some textbooks, teachers can choose to simplify them depending on the background of the students. For instance, rather than introducing the topology of the real line to students, related topological concepts can be replaced by more familiar concepts such as open and closed intervals. Some other topics such as lower and upper semicontinuity, differentiation of convex functions, and generalized differentiation of non-differentiable convex functions can be used as optional mathematical projects. In this way, the lecture notes are suitable for teaching students of different backgrounds. The second edition includes a number of improvements based on recommendations from students and colleagues and on our own experience teaching the course over the last several years. In this edition we streamlined the narrative in several sections, added more proofs, many examples worked out in detail, and numerous new exercises. In all we added over 50 examples in the main text and 100 exercises (counting parts).
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How We Got from There to Here: A Story of Real Analysis
Robert Rogers; Eugene Boman · Open SUNY · 2014
The typical introductory real analysis text starts with an analysis of the real number system and uses this to develop the definition of a limit, which is then used as a foundation for the definitions encountered thereafter. While this is certainly a reasonable approach from a logical point of view, it is not how the subject evolved, nor is it necessarily the best way to introduce students to the rigorous but highly non-intuitive definitions and proofs found in analysis. This book proposes that an effective way to motivate these definitions is to tell one of the stories (there are many) of the historical development of the subject, from its intuitive beginnings to modern rigor. The definitions and techniques are motivated by the actual difficulties encountered by the intuitive approach and are presented in their historical context. However, this is not a history of analysis book. It is an introductory analysis textbook, presented through the lens of history. As such, it does not simply insert historical snippets to supplement the material. The history is an integral part of the topic, and students are asked to solve problems that occur as they arise in their historical context. This book covers the major topics typically addressed in an introductory undergraduate course in real analysis in their historical order. Written with the student in mind, the book provides guidance for transforming an intuitive understanding into rigorous mathematical arguments. For example, in addition to more traditional problems, major theorems are often stated and a proof is outlined. The student is then asked to fill in the missing details as a homework problem.
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Discrete Mathematics
Oscar Levin · University of Northern Colorado
In the contents: 0: Introduction and Preliminaries · 0.2: Mathematical Statements · 0.E: Introduction and Preliminaries (Exercises)
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Math for Educators (MATH 010A and 010B CID120)
Lynn Mann · LibreTexts
In the contents: 1.02: Understanding Mathematical Content Knowledge · 1.04: Common Core Standards for Mathematics · 1.4.01: Kindergarten Standards Overview
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Math 375: Pre-Calculus
Roy Simpson, Sangchin Sertich & Paige Buck-Moyer · Cosumnes River College
In the contents: 01: Triangles and Circles · 1.01: Angles and Basic Geometry · 1.1.01: Resources and Key Concepts
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A Spiral Workbook for Discrete Mathematics
Harris Kwong · State University of New York at Fredonia
In the contents: 1.03: How to Read and Write Mathematics · 02: Logic · 2.02: Conjunctions and Disjunctions
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Discrete Structures
Laurie Lacey · SUNY Schenectady County Community College
In the contents: 01: Introduction to Writing Proofs in Mathematics · 1.01: Statements and Conditional Statements · 1.02: Constructing Direct Proofs
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SMC: MATH 339 Discrete Mathematics
Rohatgi · Saint Mary's College
In the contents: 0: Introduction and Preliminaries · 0.2: Mathematical Statements · 0.E: Introduction and Preliminaries (Exercises)
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MAT 310 Bridge to Advanced Mathematics
Richard Hammack · Virginia Commonwealth University
In the contents: 02: Logic · 2.01: And Or Not · 2.05: Logical Equivalence
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Elementary Foundations: An Introduction to Topics in Discrete Mathematics
Jeremy Sylvestre · University of Alberta Augustana
In the contents: 1.03: Logical Analysis · 1.04: Tautologies and contradictions · 02: Logical equivalence
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Mathematical Reasoning
Pamini Thangarajah · Mount Royal University
In the contents: 1.2: More on Logical Statements · 2.1: Subsets and Equality · 2.3: Venn Diagrams and Euler Diagrams