Types of Proofs You Might Write in Your Mathematics Classes
There are several ways to structure your argument in a proof. A few of them are elaborated upon below. Note, however, that this list is hardly comprehensive, and one should not feel limited by the ones shown here.
Direct Proofs
This is the simplest structure a proof can take. Based on our assumptions (see Writing Proofs), we hold them as true and then find other true statements our assumptions support. Therefore, structurally, direct proofs are somewhat recursive in nature, as we go from A to B by way of many smaller A’s to B’s. Through a chain or convergence of these logical connectives, we arrive to our original statement B.
Proof By Contradiction
The tenet that differentiates a proof by contradiction from a standard, direct proof is that, in addition to our assumptions from statement A, we also assume the opposite of statement B, i.e. “A is true and B is false.” From there, we show this leads to a contradiction, or something that is simultaneously true and false, which is of course impossible. The point of this is to show that, if A is true, B cannot be false and therefore is true.
Counterexamples
A counterexample can be a powerful tool. If a statement is not always true, then it holds no water. Perhaps the statement needs to be refined to a smaller case. Perhaps the statement is trying to draw the wrong conclusion. Falsehood of a statement should not be seen as a dead-end, but rather an opportunity for further exploration.
The above sections discuss writing proofs in the “always true” case. It is also important to discuss how to prove that statements are false. In this case, we provide a counterexample, a mathematical object/statement/what have you that follows all the assumptions of the overall statement but does not match the conclusion of the statement.
Breakdown into Cases
For certain proofs, it is helpful to look at specific cases separately, e.g. proving that a statement is true for all x less than 0 and then for all x greater than or equal to 0. How often this is useful varies between disciplines; it being quite common in fields such as Number Theory and Graph Theory, less so in those such as Real Analysis. There are two very important things that need to be true about your case breakdown: the first, that any possible version or iteration of statement A is covered by the cases listed (e.g. it is impossible for, say, some value for x satisfies A but does not appear under any case); the second, though less important but when satisfied makes for a much cleaner proof, that no two cases overlap (e.g. for any value of x, there is exactly one case that applies).
If one, while writing a proof by cases, finds that the proof of one case applies to both (or however many greater than one) cases, including both cases is redundant and the proof should be condensed accordingly (see “Conciseness” in “How to Write a Proof”).
Proof By Induction
This is a specific type of direct proof rather than a separate category, often used in discrete mathematics. It is used to prove something is true for all integers n. We begin by proving that the statement holds in the base case, or the lowest possible value for n. We then assume that it is true up to some arbitrary integer, say k, and then show that the statement is true in the case of k+1, establishing the statement for any integer.