Proofs & Mathematical Writing

Prerequisite path Preparation & skip guidance

Studying Proofs & Mathematical Writing

Proof-writing starts with knowing exactly what a statement allows you to assume and what it asks you to produce. Hammack separates logical language from direct proof, contradiction, counterexamples and induction, making it possible to repair one weakness at a time. Ernst takes a problem-led route in which constructing arguments is the main activity. Choose Hammack when you need explicit models of written reasoning; choose Ernst when you want sustained practice developing a claim before seeing a finished proof.

What to work on

Negate statements with nested quantifiers correctly.
Choose between direct proof, contrapositive and contradiction.
Write an induction argument with a stated base case and induction hypothesis.
Disprove a universal claim with a counterexample and explain why it qualifies.

A starting text

Book of Proof, third edition — Richard Hammack. Read sections 2.7 and 2.10 on quantifiers and negation; 4.3, 5.1 and 6.1 on proof methods; 5.3 on writing; then 9.1 and 10.1 on counterexamples and induction.

Choosing what to read

Rewrite a quantified statement in words, negate it, and test both versions on small examples. For each proof, name the domain and record where every hypothesis enters. Spend a second session revising an earlier argument for an unfamiliar reader. A correct answer with an unstated induction hypothesis or an undefined variable is a prompt for revision.

Book of Proof, third edition

Richard Hammack

Target quantifiers, counterexamples or induction without restarting an entire mathematics course.

How it reads: Explicit proof methods followed by mathematical examples.
What it expects: School algebra, elementary functions and willingness to revise prose.
What to keep in mind: Proof templates alone do not decide which mathematical idea a new problem requires.
Practice: Exercises provided; a public Solutions section starts on printed page 292. Treat it as selected solutions, not a complete answer key. Exercise sets following sections 2.10, 4.3, 5.1, 6.1 and 10.1; compare with the corresponding headings in Solutions.

Complete public PDF.

Edition, reading range and source

Read sections 2.7 and 2.10 on quantifiers and negation; 4.3, 5.1 and 6.1 on proof methods; 5.3 on writing; then 9.1 and 10.1 on counterexamples and induction.

Source · Checked 2026-10-16

An Introduction to Proof via Inquiry-Based Learning

Dana C. Ernst

Develop independence in constructing proofs and comparing classmates' arguments.

How it reads: Inquiry-based development through problems and written arguments.
What it expects: School algebra; readiness to attempt proofs before reading explanations.
What to keep in mind: The source link is a listing; the edition and available support need checking at its author route.
Practice: The catalogue describes a problem-led text; public solution coverage was not verified.

AIM listing with author/download routes; full-text access was not confirmed in the inherited record.

Edition, reading range and source

Locate the logic, sets, induction, relations and functions portions through the listed author edition; section numbering was not inherited.

Source · Checked 2026-10-11

Practice

Book of Proof exercises and selected solutions. Exercise sets following sections 2.10, 4.3, 5.1, 6.1 and 10.1; compare with the corresponding headings in Solutions.. Exercises provided; a public Solutions section starts on printed page 292. Treat it as selected solutions, not a complete answer key.

Try these before moving on

Try a proof or calculation before opening the hint. These solutions address the checks below, not the exercises in the linked textbooks.

Negate: for every real x there is a real y with y > x. Then determine the truth of the original statement and its negation.
A hint
Reverse each quantifier, then negate the inequality. For the original statement, try choosing $y$ in terms of $x$ .

Work through the solution
The original statement is $\forall x\in\mathbb R\;\exists y\in\mathbb R\;(y>x)$ . It is true: given $x$ , choose $y=x+1$ .

Its negation is $\exists x\in\mathbb R\;\forall y\in\mathbb R\;(y\le x)$ . This would assert a largest real number. It is false, since for any proposed $x$ , the choice $y=x+1$ violates the required inequality. Notice that the choice of $y$ is allowed to depend on $x$ ; swapping the quantifiers would ask a different question.
Prove that an integer n is even if and only if n squared is even, giving each implication separately.
A hint
One direction follows by writing $n=2k$ . For the other, prove the contrapositive using the form of an odd integer.

Work through the solution
If $n=2k$ for some integer $k$ , then $n^2=4k^2=2(2k^2)$ is even.

For the converse, suppose that $n$ is odd. Write $n=2k+1$ . Then

$n^2=4k^2+4k+1=2(2k^2+2k)+1,$

so $n^2$ is odd. The contrapositive gives: if $n^2$ is even, then $n$ is even. Both directions are needed for the equivalence.
Prove by induction that 1 + 3 + ... + (2n - 1) = n squared for every integer n >= 1.
A hint
After assuming the formula for $n$ , identify the next odd summand before doing any algebra.

Work through the solution
For $n=1$ , the sum is $1=1^2$ . Suppose the statement holds for an integer $n\ge1$ . The next odd summand is $2(n+1)-1=2n+1$ , so

$\sum_{k=1}^{n+1}(2k-1)=n^2+(2n+1)=(n+1)^2.$

This proves the induction step. Together with the base case, it establishes the identity for every integer $n\ge1$ . The induction hypothesis was used to replace only the first $n$ summands by $n^2$ , not to assume the desired formula at $n+1$ .

Already comfortable with this? Skip the introductory proof-method readings if you can complete all three checks, state each domain and hypothesis, and revise a proof so every inference is justified. Review quantifiers alone if negation is the remaining gap.

Review the prerequisites · Compare the full subject collection

Where this can lead

Mathematical Logic & Computability — Study formal validity and deduction once ordinary quantified arguments are readable.
Linear Algebra — Use kernels, bases and linear maps as a first setting for longer structural proofs.

Reading-source dates

Book of Proof, third edition · 2026-10-16

Reading choices revised 2026-10-17.

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Book of Proof

PDF

Richard Hammack · Textbook · Introductory

A transition from calculation to proofs, with sets, induction and cardinality. Selected solutions help self-study; it is preparation rather than an advanced subject text.

Preparation: High-school algebra.

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Additional reading

Search the complete catalogue.

Introductory reading

An Introduction to Proof via Inquiry-Based Learning

Dana C. Ernst · Textbook · Introductory · English · Full-text access unverified

A problem-led introduction to logic, sets, induction, relations and functions. Suitable for developing proof-writing by doing.

Official page / access details

Edition and sources

Checked 2026-10-11

A Gentle Introduction to the Art of Mathematics

Joseph E. Fields · Textbook · Introductory · English · Full-text access unverified

A transition to abstract mathematics through examples and proof exercises. The source lists the author text and supporting videos.

Official page / access details

Edition and sources

Mathematical Reasoning: Writing and Proof

Ted Sundstrom · Textbook · Introductory · English · Full-text access unverified

Emphasizes communicating arguments clearly, evaluating proposed proofs and learning through progress checks. Includes sets, induction and elementary number theory.

Official page / access details

Edition and sources

Version: v2.1, 2022, as listed by AIM · Checked 2026-10-11

How to Prove It: A Structured Approach

Daniel J. Velleman · Textbook · Introductory · English · Purchase / institutional access

Purchase or institutional access required; previews are not the complete book.

Introduces logical structure and techniques for reading and writing proofs. Institutional access and print availability are separate from free preview access.

Official page / access details

Edition and sources

Version: Third edition, 2019 · Checked 2026-10-11

Mathematical Discovery

Andrew M. Bruckner, Brian S. Thomson and Judith B. Bruckner · Textbook · Introductory · English · Full-text access unverified

AIM source page with an authorized full-text link.

An inquiry-based route from patterns and conjectures to justification. Read it for active exploration rather than as a catalogue of proof templates.

Preparation: School algebra; willingness to solve exploratory problems.

Use: Work through examples and exercises before advancing.

Official page / access details

Edition and sources

Version: Author edition listed by AIM; consult its title page · Checked 2026-10-11

Proofs and Fundamentals: A First Course in Abstract Mathematics

Ethan D. Bloch · Textbook · Introductory · English · Purchase / institutional access

Purchase or library subscription required; previews are not the complete book.

A transition course on mathematical reasoning, sets and functions. Use the exercises to learn how definitions drive proofs.

Preparation: School algebra; elementary functions.

Use: Work through examples and exercises before advancing.

Official page / access details

Edition and sources

Version: Second edition, 2011 · Checked 2026-10-11

How to Think Like a Mathematician

Kevin Houston · Textbook · Introductory · English · Purchase / institutional access

Purchase or library subscription required; previews are not the complete book.

Discusses mathematical reading, writing, proof construction and problem solving. A companion to first proof-based courses, not a substitute for solving their exercises.

Preparation: School mathematics.

Use: Work through examples and exercises before advancing.

Official page / access details

Edition and sources

Version: 2009; publisher online edition 2012 · Checked 2026-10-11

Proof and the Art of Mathematics

Joel David Hamkins · Textbook · Introductory · English · Purchase / institutional access

Purchase or library subscription required; previews are not the complete book.

Uses accessible mathematical problems to develop proof technique and mathematical judgement.

Preparation: School algebra and willingness to work on unfamiliar problems.

Use: Work through examples and exercises before advancing.

Official page / access details

Edition and sources

Version: 2020 · Checked 2026-10-11

How to Read and Do Proofs

Daniel Solow · Textbook · Introductory · English · Purchase / institutional access

Purchase or library subscription required; previews are not the complete book.

Organizes proof construction into explicit reasoning strategies. The official companion offers supporting material, not the complete textbook.

Preparation: School algebra; introductory sets.

Use: Work through examples and exercises before advancing.

Official page / access details

Edition and sources

Version: Fourth-edition companion site; consult publisher for later editions · Checked 2026-10-11

Proofs and Concepts: The Fundamentals of Abstract Mathematics

Dave Witte Morris and Joy Morris, incorporating material by P. D. Magnus · Textbook · Introductory · English · PDF

Author's Deductive Press page links the full 195-page PDF under CC BY-NC-SA 2.0.

A bridge course connecting elementary logic with sets, functions, equivalence relations, induction and cardinality.

Preparation: School algebra; basic mathematical notation.

Use: Practice complete arguments before entering proof-based algebra and analysis.

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Edition and sources

Version: Preliminary version 0.95, January 2025 · Checked 2026-10-13

Mathematical Method

Jesper Lützen and Ian Kiming · Course notes · Introductory · Danish · PDF

Complete 199-page course book opened from the department's official archive.

English title: Mathematical Method. Introduces logic, proof strategies, induction and the analysis–synthesis approach to mathematical problem solving.

Preparation: School algebra and geometry; basic mathematical notation.

Use: Practise writing proofs and analysing problems before taking abstract undergraduate courses.

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Edition and sources

Original title: Matematisk Metode

Version: October 17, 2008, as printed on the title page · Checked 2026-10-13

Introduction to Abstract Mathematics

Flemming Topsøe · Course notes · Introductory · Danish · PDF

University department archive provides the complete 164-page Matematik Y text.

English title: Introduction to Abstract Mathematics. Builds a foundation in propositional and predicate logic, sets, functions, orders and induction, with exercises and challenges.

Preparation: School mathematics; willingness to read formal definitions.

Use: Prepare for proof-based university mathematics and explore elementary foundations.

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Edition and sources

Original title: Introduktion til abstrakt matematik

Version: 2002 edition · Checked 2026-10-13

Mathematical Discussions

Timothy Gowers · Expository essay collection · Introductory · English · Online text

Full text on the author’s website or archive.

Dialogues and explanations explore why definitions and proofs arise in algebra, analysis, geometry and logic.

Preparation: School mathematics; willingness to work through proofs.

Use: Supplement undergraduate courses with conceptual explanations.

Open online text / archive

Edition and sources

Version: Author’s online collection; individual essays have their own histories · Checked 2026-10-13

Does One Have to Be a Genius to Do Maths?

Terence Tao · Mathematical essay · Introductory · English · Online text

Full text on the author’s website or archive.

Discusses the roles of learning, persistence and cooperation in mathematical research.

Preparation: No university mathematics required.

Use: Read a personal perspective on developing mathematical ability.

Open online text / archive

Edition and sources

Version: Author’s online essay; current page checked October 2026 · Checked 2026-10-13

A Mathematician’s Lament — Original Essay

Paul Lockhart · Mathematical-education essay · Introductory · English · PDF

Free PDF at the author or authorized archive.

Lockhart argues for imagination and discovery in school mathematics through a deliberately polemical essay.

Preparation: No university mathematics required.

Use: Compare a teacher’s viewpoint with your own learning experiences.

Open full PDF

Edition and sources

Version: Original 25-page essay, 2002; separate from the expanded 2009 book · Checked 2026-10-13

Intermediate reading

An Infinitely Large Napkin

Evan Chen · Unfinished book draft · Intermediate · English · PDF

A broad bridge from undergraduate subjects toward first-year graduate ideas. Some planned chapters and exercises are unfinished; it is not a substitute for a full course in each field.

Preparation: Comfort with proofs; undergraduate algebra and analysis.

Use: Orient a personal reading programme across subjects.

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Edition and sources

Version: Version 1.6; evolving draft, source page updated April 2026 · Checked 2026-10-11

Proofs from THE BOOK

Martin Aigner and Günter M. Ziegler · Textbook · Intermediate · English · Purchase / institutional access

Purchase or library subscription required; previews are not the complete book.

A collection of elegant arguments across several fields. Read after learning basic proof techniques; it is not a systematic first curriculum.

Preparation: Proof-writing; elementary algebra and combinatorics.

Official page / access details

Edition and sources

Version: Sixth edition, 2018 · Checked 2026-10-11

An Introduction to Proofs and the Mathematical Vernacular

Martin V. Day · Textbook · Intermediate · English · PDF

Author's Virginia Tech page provides the full 147-page book, CC BY-NC-ND 3.0 US.

Uses number systems, polynomials and determinants to practise reading and constructing longer mathematical arguments.

Preparation: Calculus sequence; basic matrix computations.

Use: Bridge computational courses to upper-level mathematical writing.

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Edition and sources

Version: December 7, 2016 · Checked 2026-10-13

The Two Cultures of Mathematics

Timothy Gowers · Mathematical essay · Intermediate · English · PDF

Free PDF at the author or authorized archive.

Gowers discusses theory-building and problem-solving, using combinatorics to examine mathematical value and communication.

Preparation: Some undergraduate mathematics helps with examples.

Use: Read a mathematician’s perspective on different research styles.

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Edition and sources

Version: Author manuscript for Mathematics: Frontiers and Perspectives (2000) · Checked 2026-10-13

On Proof and Progress in Mathematics

William P. Thurston · Mathematical essay · Intermediate · English · PDF

Free PDF at the author or authorized archive.

Thurston reflects on understanding, proof and communication, drawing on foliations and three-dimensional geometry.

Preparation: General interest in mathematics; topology helps with examples.

Use: Compare formal proof with broader mathematical understanding.

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Edition and sources

Version: 1994 author preprint; Bulletin of the AMS 30 · Checked 2026-10-13

What Is Good Mathematics?

Terence Tao · Mathematical essay · Intermediate · English · PDF

Free PDF at the author or authorized archive.

Examines multiple sources of mathematical value and the connections that can develop between them.

Preparation: Some experience with undergraduate mathematics.

Use: Reflect on mathematical taste and choices of research problems.

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Edition and sources

Version: 2007 author preprint · Checked 2026-10-13

Mathematics in the Age of AI

Terence Tao · Public-lecture essay · Intermediate · English · PDF

Free PDF at the author or authorized archive.

Tao considers how mathematical practice and collaboration might respond to research-capable AI tools.

Preparation: General mathematical interest.

Use: Explore one researcher’s perspective on emerging tools.

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Edition and sources

Version: 2026 ICM public-lecture essay · Checked 2026-10-13

On Writing

Terence Tao · Expository guidance collection · Intermediate · English · Online text

Full text on the author’s website or archive.

Advice and linked essays address mathematical exposition, revision and communication with readers.

Preparation: Experience reading mathematical arguments.

Use: Develop clearer mathematical writing.

Open online text / archive

Edition and sources

Version: Author’s online collection · Checked 2026-10-13

Science and Hypothesis

Henri Poincaré; translated by W. J. Greenstreet · Mathematical philosophy / popular science · Intermediate · English · PDF

Complete English-translation PDF at Project Gutenberg.

Discusses mathematical reasoning, geometry, hypotheses and physical theories from Poincaré’s historical viewpoint.

Preparation: Elementary mathematics; interest in scientific philosophy.

Use: Read an influential historical discussion of mathematical knowledge.

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Edition and sources

Original title: La Science et l’Hypothèse

Version: 1905 English translation; Project Gutenberg PDF · Checked 2026-10-13