Carrying out math proofs can be one of the hardest things for students to do. Undergraduates in math, computer science, or other related fields will likely encounter proofs at some point. By simply following a few guidelines you can clear the doubt about the validity of your proof.
Steps
Step 1. Understand that mathematics uses information you already know, especially axioms or the results of other theorems
Step 2. Write down what is given, as well as what you need to prove
It means that you have to start with what you have, use other axioms, theorems or calculations that you already know are true to arrive at what you want to prove. To understand well you need to be able to repeat and paraphrase the problem in at least 3 different ways: by pure symbols, with flowcharts and using words.
Step 3. Ask yourself questions as you go
Why is this so? and Is there a way to make this fake? are good questions for any statement or request. These questions will be asked by your teacher at each step, and if you can't check one, your grade will drop. Support every logical step with a motivation! Justify your process.
Step 4. Make sure the demonstration happens at every single step
There is a need to move from one logical statement to another, with the support of each step, so that there is no reason to doubt the validity of the proof. It should be a constructionist process, like building a house: orderly, systematic and with properly regulated progress. There is a graphic proof of the Pythagorean theorem, which is based on a simple procedure [1].
Step 5. Ask your teacher or classmate if you have any questions
It's good to ask questions every now and then. It is the learning process that requires it. Remember: there are no stupid questions.
Step 6. Decide on the end of the demonstration
There are several ways to do this:
- C. V. D., that is, as we wanted to prove. Q. E. D., quod erat demonstrandum, in Latin, stands for what had to be proved. Technically, it is only appropriate when the last statement of the proof is itself the proposition to prove.
- A bullet, a filled square at the end of the proof.
- R. A. A (reductio ad absurdum, translated as to bring back the absurd) is for indirect demonstrations or for contradiction. If the proof is incorrect, however, these acronyms are bad news for your vote.
- If you are not sure if the proof is correct, just write a few sentences explaining your conclusion and why it is significant. If you use any of the above acronyms and get the proof wrong, your grade will suffer.
Step 7. Remember the definitions you have been given
Review your notes and book to see if the definition is correct.
Step 8. Take some time to reflect on the demonstration
The goal was not the test, but the learning. If you just do the demonstration and then go further, you are missing out on half of the learning experience. Think about it. Will you be satisfied with this?
Advice
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Try to apply the proof to a case where it should fail and see if it actually is. For example, here is a possible proof that the square root of a number (meaning any number) tends to infinity, when that number tends to infinity.
For all n positives, the square root of n + 1 is greater than the square root of n
So if this is true, when n increases, the square root also increases; and when n tends to infinity, its square root tends to infinity for all ns. (It might seem correct at first glance.)
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- But, even if the statement you try to prove is true, the inference is false. This proof should apply equally well to the arctangent of n as it does to the square root of n. Arctan of n + 1 is always greater than arctan of n for all n positives. But arctan does not tend to infinity, it tends to laziness / 2.
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Instead, let's demonstrate it as follows. To prove that something tends to infinity, we need that, for all numbers M, there exists a number N such that, for every n greater than N, the square root of n is greater than M. There is such a number - is M ^ 2.
This example also shows that you need to carefully check the definition of what you are trying to prove
- Proofs are difficult to learn to write. A great way to learn them is to study related theorems and how they are proved.
- A good mathematical proof makes each step really obvious. High-sounding phrases might earn marks in other subjects, but in mathematics they tend to hide gaps in reasoning.
- What looks like failure, but is more than what you started with, is actually progress. Can give information on the solution.
- Realize that a proof is only good reasoning with each step justified. You can see around 50 of them online.
- The best thing about most proofs: they have already been proven, which means they are usually true! If you come to a conclusion that is different from what you should prove, then it is more than likely that you are stuck somewhere. Just go back and carefully review each step.
- There are thousands of heuristic methods or good ideas to try. Polya's book has two parts: a “how to do if” and an encyclopedia of heuristics.
- Writing a lot of proofs for your demonstrations isn't that uncommon. Considering that some assignments will consist of 10 pages or more, you'll want to make sure you get it right.
