Proof Techniques

All mathematical proofs are deductive. A deductive argument is an argument that is a series of implications, in which the conclusion of each argument is used as at least part of the hypothesis for the following implications. The argument is valid if and only if each implication is valid. In order for the conclusion to be true, the initial hypothesis must be true as well.

One method of proof is called a direct proof. A direct proof involves establishing the truth of a proposition by making it the conclusion of a deductive argument whose initial hypothesis is taken as true.

Another method of proof is known as an indirect proof. It is based around fundamental laws of logic that state that, given any proposition, either the proposition is true or the negation of the proposition is true, but not both. So, if you show that the negation of a proposition has to be false, then the original proposition has to be true.

Mathematical induction can be used to prove various theorems about the natural numbers. The principle of mathematical induction, in a nutshell, states that, if a statement holds for the number 1, and the statement holding for the number n implies that it holds for the number n + 1, then the statement is true for all of the natural numbers.

Sources used (see bibliography page for titles corresponding to numbers): 51.