Various methods of proof

 Direct Proofs

In general, a theorem is a statement of the form p →q. The hypothesis p will likely be a compound statement, and some of its components might not even be explicitly stated. At the foundational level, our task in writing the proof of a theorem is to show that p → q is a tautology. However, the final product that is called a proof will be in sentence form, and not look anything like the manipulation of a series of symbols. In our first example, we employ the method of direct proof.

Proof by Contradiction

Direct proofs and are effectively an effort to show that p →q is a tautology. Sometimes, however, it’s easier to prove p →q is a tautology by showing￢(p →q)is a contradiction. In order to construct a proof by contradiction, we first have to recall the negation of p → q. It is also helpful to remember that an ifthen statement might sometimes be better stated using ￢and∨.

Proof by Induction

Induction is a very powerful technique used regularly in Mathematics. It may appear misleading because it sometimes appears like we assume what is to be proved. The advantage of this method is that it is easy to recognise when to use, and also how to use and only two conditions are needed to be checked to apply it.

Proof Using the Contrapositive Method

The technique of proving by using the contrapositive method uses the fact that the statement ‘p→ q’ is equivalent to ‘not q → not p’. The contrapositive of the statement p→ q is ¬q → ¬p, as we have already seen. The contrapositive method is the use of this equivalence. It is an indirect method, wherein to prove ‘p → q’we start with ¬q(and proceed to show that not p is true).

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