If you are familiar with Polya’s How to Solve It, one of the most well-known classic books in mathematical problem solving, a similar book exists for learning mathematical proofs. Daniel Velleman’s How to Prove It: A Structured Approach is one of the good books available for learning the structure of proofs.
The books include topics onm Sentential logic, Quantification Logic, Proof Strategies, Relations, Functions, Mathematical Induction and Infinite Sets. It contains detailed explanation and numerous examples on different types of proofs and the logic behind them. It contains explanations on connectives, quantifiers, truth tables, countable and uncountable sets and more.
How to Prove It is a recommended book for undergraduate mathematics students as well as advanced high school students who plan to be mathematics majors.
I have written quite a number of articles on mathematical proofs, so I want to summarize them in this post. Most of these proofs are high school level, so students who are mathematically inclined are encouraged to read them.
I will update this list every time I have written proof-related posts, so you may want to book mark this post.
Two years ago, I worked as part-time geometry instructor in a technical school near our university. Most of the students in that school were quite clueless about the notion of proofs, so I tried to find ways to introduce proofs in an intuitive manner.
One lesson I developed was on proving that a quadrilateral formed from paper folding is a square. I let the students create a square from a piece of bond paper without using any measuring instrument; only folding and cutting were allowed.
As expected, most of the students used the method shown in the figure above. For the sake of discussion, we label the corners and critical points of the bond paper. Most of the students constructed the square using the following steps (see figure): » Read more