How do you do mathematical proof?

Answer 1

Proving statements can be challenging if you are not used to know some math definitions and forms. This is the pre-requisites of proving things! Math maturity is the plus. Math maturity is the term that describes the mixture of mathematical experience and insight that can't be learned. If you have some feelings of understanding the theorems and proofs, you will be able to work out the proof by yourself!

Formulating a proof all depends on the statement given, though the steps of proving statements are usually the same. Here, I list some parts in formulating the proof in terms of general length of the proof.

"Let/assume [something something]. Prove that [something something]"

1. Read the whole statement several times.
2. Start off with what is given for the problem. You can write "we want to show that [something something]"
3. Apply the definitions/lemmas/theorems for the given. Try not to skip steps when proving things. Proving by intuition is considered to be the example of this step.

"Let/assume [something something]. If [something something], then show/prove [something something]"

The steps for proving that type of statement are similar to the ones above it.

"Prove that [something something] if and only if (iff) [something something]"

This can sometimes be tricky when you prove this type of statement. That is because the steps of proving statements are not always irreversible or interchangeable.

To prove that type of statement, you need to prove the converse and the conditional of the statement.

When proving the conditional statement, you are proving "if [something something], then [something something]". To understand which direction you are proving, indicate the arrow. For instance, ← means that you are proving the given statement on the right to the left, which is needed to be proved.

When proving the converse statement, you switch the method of proving the whole statement. This means that you are proving the given statement from "left" to "right". Symbolically, you are proving this way: →.

Note: Difficulty varies, depending on your mathematical experience and how well you can understand the problem.

Another Note: If you fail in proof, then try again! Have the instructor to show you how to approach the proof. Think of proving things as doing computation of numbers! They are related to each other because they deal with steps needed to be taken to prove the statement.