In order to really dive into higher or advanced mathematics it is essential to have a solid foundation in how to read and write proofs. This foundation is normally built up in a course called “Introduction to Proofs” or something similar and is an important rite of passage for any person studying mathematics. Prior to this point much of mathematics is presented in a formulaic manner: if you see this type of problem use this type of formula/technique to solve it. If any explanation is given it is often ‘hand-wavy’ or lacking detail. After taking a proofs course students move into higher level math where the focus is on theory and rigor. The shift in focus can be jarring to some students, which is why it is so important to have a solid foundation in proofs before moving on to more advanced mathematics. In this post I will lay out a road map for studying the basics of proofs, sets, and logic that is particularly geared towards the self-learner who is studying on their own.
Some schools combine their introduction to proofs class with another subject rather than have it be a standalone class. For example, some schools have “Real Analysis with an Introduction to Proof” as a course, or use a course in discrete math as a place to teach proofs. While this is a viable way to go about things, I personally prefer the idea of taking time to focus specifically on proofs without having to worry about rushing onto another subject. Learning proofs is a large enough undertaking on its own.
The biggest disadvantage self-learners face when learning proofs on their own is the lack of feedback. In a classroom setting there will be classmates to study with and a professor to critique your final work, but as a self-learner you do not really have access to that same level of interaction. It is possible to get some feedback from online communities and forums, but not to the same extent that a professor can provide. The best remedy to this is to read a lot of proofs. The more proofs you read, the more you will start to recognize the difference between a good proof and a bad one and be able to apply those lessons to your own work. In support of this, I will give a number of recommended books to look through. The other thing that self-learners must do is make a habit of critiquing their own work. After you complete a rough draft for a proof look it over, revise it, and then repeat until you cannot find a way to improve it any further.
Goals for the Course:
The end goal for studying basic proofs is to be comfortable reading and writing mathematical proofs, as well as using and understanding basic mathematical notations. The secondary goal is to ease the transition from lower-level mathematics to more advanced mathematics.
What Material Should Be Covered?
Typically an intro to proofs course begins with a discussion of logic and logical statements. This includes statements involving ‘if’, ‘if and only if’, ‘and’, and ‘or’. After this, the basics of proofs are introduced such as direct proofs, induction, and proofs by contradiction. The next topic is normally the basics of set theory, including intersections, unions, and subsets. At this point, students begin doing some proofs involving sets. An example might be to prove that two sets are the same, or that one is a subset of the other, or that their intersection is empty.
Next, the concept of a function is presented. A considerable amount of time is normally spent on functions since they are the fundamental building blocks to many areas of mathematics. The topics include one-to-one and onto functions, bijections, inverses, and the compositions of functions. My proofs professor drilled these topics into my head until I could practically do these proofs in my sleep. Functions are followed up by lessons on operations, relations, and partitions.
Lastly, most proofs courses spend some time on what might be called ‘applications’, for lack of a better word. Students will be asked to prove some basic results from topics like real analysis, number theory, discrete math, or abstract algebra. The exact topics covered differ from professor to professor and from book to book. This might include proving statements like “there are infinitely many prime numbers” or “\(n^2\) is equal to the sum of the first \(n\) odd numbers”. Students may work wit sequences and proving the convergence of limits. Number theory in particular works really well for this sort of thing since many basic results in number theory are easy to prove but help build good proof writing techniques.
Which Books Should Be Used?
Luckily, there are a number of good quality resources for learning about proofs. I will start by recommending three freely available books and then later recommend some books which you might consider purchasing if you want additional resources.
First up, the “Book of Proof” by Richard Hammack (available free here). This is a commonly recommended textbook and for good reason. The writing is clear and involves a lot of worked examples from various areas of mathematics. Every section of the book has a set of exercises with it and the solutions to the odd-numbered exercises are in the back of the book. What is excellent about these solutions is that they include full proofs, an invaluable resource for self-learners. The book has very little in the way of unnecessary fluff in it. The author notes that chapter 3 can be skipped, but if you are already reading the rest of the book you might as well read chapter 3, too.
This book follows the typical approach to proofs where the student is asked to forget, temporarily, most of their previous knowledge in mathematics in order to build a new foundation from scratch. The upside of this approach is that the book has very few prerequisites. A student should have a solid foundation in basic algebra.
The book recommends a fourteen week course of study but for self-study I would expect anywhere from 2 – 8 months.
The next book I would recommend is “An Introduction to Proofs and the Mathematical Vernacular” by Martin V. Day (available free here). This book is shorter than the Book of Proof and it takes a different stylistic approach. While many proofs courses ask the reader to set aside their previous knowledge, this book specifically does the opposite. The book assumes a solid knowledge of calculus and some linear algebra, even going so far as to introduce some less commonly taught concepts results in calculus as a training ground for working with proofs.
Two downsides with this book are the pacing and the minimal amount of problems/solutions. The book moves quickly from one topic to the next, and while I like how the book presents things it might work better as a sort of second exposure to proofs. Although the book has a number of worked examples and some problems at the end of sections, it has no solutions. Solutions are absolutely essential when learning proofs since they may be the only form of feedback available to the self-learner.
Overall, I recommend this book to a reader looking for a challenge or who already has some mathematical exposure.
The last free book I would like to recommend is “Mathematical Reasoning, Writing, and Proof” by Ted Sundstrom (available free here). This book is very long and detailed, holding the reader’s hand through every step of the process. It is a great reference textbook when reading another textbook, or a source for problems since it has solutions to the odd-numbered exercises at the end of the book. The writing is extremely clear and I feel the book has good pacing. It follows a similar progression of topics as the Book of Proof does, with only a few minor differences.
Some other notable proofs books include:
- “Introduction to Mathematical Thinking” by Keith Devlin
- “How to Prove It: A Structured Approach” by Daniel J. Velleman
Some non-proofs books that are approachable and still provide good practice include:
- “Introduction to Graph Theory” by Richard J Trudeau
- “A Book of Abstract Algebra” by Charles C. Pinter
- “An Introduction to Number Theory with Cryptography” by James Kraft and Lawrence Washington
Tips to Make Learning Easier:
As far as areas of mathematics go, introductory proofs is a fairly straightforward area to study. The material is standardized and there isn’t much fluff. That said, it is not easy material. Here are some tips to make your journey easier:
- Don’t get bogged down in notation. Theoretically, everything in mathematics can be written out entirely in words. The reason we use notation is because words can get cumbersome. However, if you are struggling, try rewriting what you are working on using only words.
- If you can not prove something, try to disprove it instead. Coming at a problem from a different mindset could help you gain insights that will help you solve the original problem.
- When in doubt, go back to the definitions.
- Before writing a formal proof, try to convince yourself that a statement is true. Allow yourself to use informal reasoning and intuition to look for a justification. After you have convinced yourself you can translate your thinking back into formal reasoning.
- You may need to spend some time thinking about problems. Some problems may need to sit in your brain for a few days before you find the solution. You should allow yourself to have this time and not cheat yourself by prematurely looking at the solutions.
- Doing multiple drafts of proofs is important and a good way to grow.
- Communicate with others and practice reading and writing mathematics.
Thank you for taking the time to read this post. I sincerely hope that this has helped you on your mathematics learning journey, and specifically helped you to learn about basic proofs. This is my first ‘curriculum’ post but I will be writing more in the coming days and weeks. Please leave your feedback in the comments here or contact me on Twitter.
Great article! Thanks for sharing