# How To Prove It - Conclusion and Review

This is the first time after college(2007) I completed a book cover to cover while attempting all of its exercises.

I started learning mathematics with a book Linear Algebra and as soon as I reached towards its end, I had already forgotten many concepts from the initial chapters.

Soon I realised the need for a change in strategy in learning mathematics.

As I was wandering online, I came to know about How to Prove It from these two excellent posts:

This book was not a part of my initial plan but after reading above posts and many reviews, I set out for taking this endeavour with following goals:

• Brush up my mathematics skills.
• Learn how to write proofs.
• Attempting all the exercises(for better understanding and remembering).
• To feel good about reading a mathematics book cover to cover :)

I also found a blog My Technical Scratch Pad containing solutions of exercises from this book. It motivated me further in my endeavour and helped me a lot in many solutions.

It has roughly taken about 450-500 hours or 3.5-4 months(40 hrs a week), much longer than I had initially anticipated. Many times it happened that one problem ended up consuming most of the day. Also about 20-25% of the time was spent in putting these solutions online.

Here is how I found the book.

Author does not expect much from the reader and begins with very basic concepts and slowly progresses towards quantifiers, then set theory, relation and functions, mathematical induction and finally, infinite sets.

Inside introduction, author gives proof of few theorems in an intuitive way. Later when armed with all the proofing techniques all of those proofs were revisited and reader can clearly see the difference in his understanding for reading and writing proofs.

All the techniques of proofs(except induction) are covered in chapter-3. Post that, book introduced other topics like relations and functions and employs proof techniques for proving theorem in these topics. It was a great way to demonstrate that techniques learned for writing proofs are independent of any area and can be applied anywhere in mathematics.

I loved the treatment of proof by contradiction and mathematical induction. Cracking the corresponding exercises was a very rewarding experience. In many proofs when no approach seems to be working, proof by contradiction comes to the rescue. Similarly power of proof by induction was on display in solving many humongous problems.

All exercises were ordered from easy to moderate preparing the reader along the way to learn writing proofs for easier to challenging ones. Many exercises are built on top of the theorems from earlier exercises. This is a good thing as it helped me in two ways: revising the older chapters and discovering errors in my proofs.

There were many exercises asking the reader if the given proof is correct. Many times proof looked correct but turned out wrong because of a conceptual mistake. This helped tremendously in clearing many misconceptions.

In most of the sections, author also explains about how he arrived at a solution which helped in understanding how to approach a problem.

Finally in the last chapter author picked up a relatively advanced topic and employs all the proof techniques learned. In this chapter author does not go into explaining the proof structure but writes in a mathematical rigour so that reader should be able to read those proofs and gets an overall idea about reading and writing proofs by giving more focus to the topic than the proof technique.

One small thing that could have been better is the treatment of empty sets. I got confused while solving many exercises and felt like missing on some concepts regarding empty sets specially while dealing with family of sets.

To summarise,

• Quantifiers are everywhere.