Specific Tips on Problem-Solving

  • Start by identifying the conclusion you are supposed to reach, and any information you are given. Try writing the hypotheses at the top of your paper and the conclusion at the end, and then think about what steps could go in the middle. Think about what kind of approach might be able to get you from the beginning (your hypotheses) to the end (your conclusion). If there is some idea that sticks out, explore it. If you have several hypotheses, think about how they might work together.

  • Look at your notes (or the book) and try to identify anything that looks similar or might be helpful. (The textbook exercises are usually placed in that section for a good reason: you will probably need ideas or results from that section to solve them!) If you are asked to prove a result similar to something whose proof you already have, try seeing if you can adapt that method to your problem.

  • If you’re trying to prove something directly and getting stuck, try arguing by contradiction instead, and see what can go wrong. If you want to prove that all As are B, then examine what happens if you had an A that was not B: having an extra few properties (namely, “not B”) lying around might give you more ideas.

  • If you are trying to prove that two statements are equivalent (A if and only if B), try splitting it into two conditionals: if A then B, and if B then A. Each conditional gives you something to start with along with a goal you want to reach.

  • When trying to prove that two expressions are equal, try starting with one side of the expression and doing transformations on it until you obtain the quantity on the other side.

  • If you’re confused about what the hypotheses or the conclusion say, make sure to translate the problem into concrete information (the “definition chase”). Instead of trying to manipulate a statement like “\(S\) spans the vector space \(V\)”, write out the statement in a usable form: for any vector \(w\) in \(V\), there exist vectors \(v_1, \, v_2 , \dots , v_n\) in \(S\) and scalars \(a_1 , \, a_2 , \dots , a_n\) such that \(w = a_1 v_1 + \cdots + a_n v_n\). This will give you something you can actually work with.

  • If you have to show something is true for every integer \(n\), try seeing if you can do it by induction. Using induction often seems a little like magic because it converts a difficult direct argument into a few smaller ones, each with clear instructions.

  • (For induction problems) When writing an induction proof, make sure to label the parts of the proof (“base case” and “inductive step”), and also indicate clearly what variable you are inducting on. Each of these pieces has a clear goal: the base case is usually an easy example where the result is obvious or almost obvious, while the inductive step gives you a hypothesis to start with and a goal to reach. Also keep in mind that you can always use “strong induction” (that is, assume all of the previous cases hold, rather than just the immediately previous one).

  • If a problem has multiple parts, they are often related in some way. Especially in problems that ask you to prove things, always be on the lookout for a way to use the results from earlier parts of the problem in the later parts. For example, if part (a) of a problem asks you to prove that \(f(x) = x^3 - x\) whenever \(x < 2\), and then part (b) asks you to find \(f(1)\), you can simply say “by part (a), we immediately see \(f(1) = 0\)”.

Specific Tips on Writing Up Proofs.

Solving a problem is not the same as writing a proof. Once you have figured something out, you still have to write it up.

  • A proof is a sequence of statements, written in sentences, that establish a conclusion starting from some hypotheses. A proof is not the same as writing up the details of a calculation (e.g., how you computed a derivative), nor is it merely a succession of equalities: it is a sequence of statements written using words and symbols (or possibly just words).

  • You should be able to read a proof aloud, at least in principle, and have it make perfect sense. This includes the equations you have written, as mathematical symbols and equations have verbal translations: the equation \(x^{2}=9\) can be translated into the equivalent verbal statement “\(x\)-squared equals nine”.

  • Standards for good writing also apply to proof-writing: you should use correct grammar, write in full sentences, organize sentences into paragraphs, etc. If the proof is hard to read (e.g., with lines erased or scribbled out, or missing parts added in separate areas of the page), you should rewrite it cleanly.

  • Proofs should contain no gaps. Every statement in a proof should follow logically, in an obvious way, from previous statements or known facts. The definition of “in an obvious way” is a little bit subjective, but as a general rule, if your argument would not completely convince another MTH 235 student who hasn’t seen that particular problem, it needs more detail.

  • A well-written proof should be easy to read and easy to follow: if there are multiple parts to an argument, they should be clearly separated from each other.

  • It is good style to be very clear, at the very beginning, about the structure of the proof: state immediately what result you are proving and (if you are using a special proof style like proof by induction or proof by contradiction) what kind of proof you are using.

  • There are hundreds of examples of properly-written proofs in the textbook, and you will see dozens more in lecture. Most of these proofs are longer and more complicated than the ones you are expected to generate, but you should aspire to write proofs that are similar to ours in style and form.

External Guides

  • Constructing and Writing Proofs in Mathematics: Extensive guide (an actual textbook) on the different methods when solving a problem and writing up a proof.

  • A Primer on Mathematical Proof: A more concise guide on the most common techniques and tips when writing a proof.

  • Proof-Writing Tips: This is a short list of 10 tips on writing proofs, with advice both about how to interpret a proof problem and how to format and style your writeup.

  • Introduction to Mathematical Arguments: This is a much lengthier document giving a comparatively gentle introduction to mathematical proofs. If you are in need of something to read to help you understand how to start writing proofs, start here.

  • Some Remarks on Writing Mathematical Proofs: Another list of suggestions on writing up your proofs, similar to the ones on this page. Also contains more remarks about how to format mathematics when you are writing it.