Why do we emphasize so much on writing proofs? Many mathematical ideas, like scientific discoveries, need to be reviewed to double check the work. In fact, not all research has turned out to be correct. Mistakes do lurk around. However, if the work is not clear, then the result could be deemed dubious. Hence, it is important to develop a clear, if not professional, manner of writing a proof.
How to write a proof
A proof is a set of logical bridges leading from known results (premises) to the conclusion. Here are the "to be followed" principles:
Start by writing what you want to show, mathematically.
Example: "We would like to show that n(n+1)/2 is a factor of 1^k+2^k+...+n^k, where n is a natural number and k is an odd positive integer."
Define all variables as they are to be used. Be clear and concise.
Example: "Let x represent the number of vertices of graph G with degree 3, and y represent the number of vertices of graph G with degree 4."
Tip: Use "respectively" to help shorten your definition. "Let x and y represent the number of vertices of graph G with degree 3 and 4 respectively."
Note: Sometimes, some problems require that information of a variable be stated, such as "The problem is equivalent to showing that 2^x>x^2, where x is an integer greater than 5."
State all nontrivial (use discretion when necessary) premises needed to reach the subconclusion
Example: "Since xy+yz+zx-1 is a multiple of x, y and z, and that x, y and z are coprime, thus xy+yz+zx-1 is a multiple of xyz."
To quote a known result, write out the corresponding statement fully, then show how it applies to your proof. If you know the name of the result, do state it.
Example: "By Fermat's Little Theorem, a^p is congruent to a modulo p. Hence, ..."
Note: You may not need to state everything, just make sure that the relevant required information is in. Sometimes, if the name of the theorem is known, you can write: "By Fermat's Little Theorem, (subconclusion)"
Number all subconclusions that will be used again later in the proof. This will be useful for referencing for the reader.
Example: "Hence, m is a multiple of 3 --- 1...Hence, m is prime --- 2. By subconclusion 1, m is a multiple of 3. However, by subconclusion 2, m is prime, so m=3."
Use indents. Subconclusions to the leftmost, indent the rest of the working.
Do not skip steps. If something is obvious, state the premises that lead to it. Always check that the premises that you presented are sufficient to lead to the subconclusion. If not, add in the other premise or the missing link.*
State out the final conclusion in the last step, which is what you set out to prove.
Example: "Therefore, there are only n! ways to arrange n different objects."
We hope this has helped you! After all, the ONLY point of writing proofs is such that you and other people can check your work easily. We know proof writing is tedious, and you will experience it soon enough. Do not lose your determination! It is really necessary to improve on your thinking.
Time to embark on your mathematical journey!
*This is usually the step most people have difficulty in. Always refine the proof by looking back at the logic tree, and check for necessity and sufficiency.
