Strategies for Proofs

Good practices

If we have a statement at hand and are looking for a method to prove it, we must know there is no procedure to develop proof. There can be more than one way to prove that a mathematical statement is correct. It is more of an art form to construct a valid mathematical argument for establishing the truth value of a well-posed mathematical question. The question for a method to develop theorems is the same as the question for a technique to create good poetry. Yet, we can get inspiration from others’ work and learn the basic principles to start our journey to construct proofs. In this regard, the best strategy is to read and understand the classical proofs, which is like getting inspiration for your art from the work of famous classical artists. Certain practices can help while developing an argument.

• Formalize the question concisely and clearly. Half of the solution lies in a good understanding of the question.
• Try using different methods, as we do not know which will work, but we will undoubtedly get insight into the problem.
• Keep all the possibilities open; maybe the statement we are trying to prove true is false in reality.
• Check the correctness of the argument multiple times by writing it down cleanly and in detail.

Types of proof

If we look at the existing proofs, there is an excellent use of specific tools and types of arguments. The scope of this course does not allow us to cover all of them. However, we can list down a few names to help further exploration:

• Direct proof
• Proof by contrapositive
• Proof by contradiction
• Principal of mathematical induction
• Existence proof
• Proof by construction
• Proof by cases
• Exhaustive proof
• Uniqueness proof
• Searching counterexample
• Combinatorial proof

The list presented above is not exhaustive in any fashion. It is just a list of basic proof methods. The content we have learned in this course should provide sufficient ground to understand the above-listed techniques.