# Methods of Proof

To prove mathematical results, in general we use any of the following methods.

**(1)** When statements of the form *p*** q** are used to arrive at the truth of a certain mathematical result, this kind of approach to establish the result is known as the “

**direct proof**”.

**(2)** In case an equivalent statement is used to arrive at the result, then this method of proof is known as “**indirect proof**”. In particular, in place of arriving at a statement ** q** form a statement

**if we arrive at**

*p***from**

*p***then such a kind of proof is called “**

*q***contrapositive proof**”.

Symbolically, we establish *q*** p** in place of

*p***.**

*q***(3)** Sometimes, to prove a certain result ** p**, it is convenient to prove the equivalent form (

**), i.e. negation of**

*p***. This method of proof involves**

*p***contradiction of negation**of

**.**

*p*For example, to prove the uniqueness of a certain element , if possible, suppose is

an element satisfying the properties of then we show that . In such a case, instead of directly proving the proposition ** p= **“ is unique,” i.e. for every satisfying properties of , , we prove (

**) or () i.e. .**

*p***(4)** To prove a mathematical result ** p(n)** concerning natural number n, it is sometimes convenient to prove the truth of

**for**

*p(n)***whenever**

*n +**1***holds true for**

*p(n)***. And if**

*n***is the least natural number for which**

*m***holds, then the truth of**

*p(n)***is established for all .**

*p(n)***Such a method of proof is called the “**

**method or principle of finite induction**”. This method depends upon the fact that the natural numbers follow the principle of finite induction.

To illustrate this method, let us prove for all natural numbers that

The result (*) is evident for . Let us suppose that it is true for some . Then (*) gives that

And

i.e.

Therefore,

i.e.

Thus, (*) being true for is also true for. And it already holds for . Hence (*) is established.

The use of methods described in **(1) **to **(4) **will appear in various places in our subsequent course of study. In various types of proofs the **quantifier symbols** and are found to be quite useful. These symbols and stand for the words “**for all**” and “**there exists**,” respectively. Whenever we write, it stands in the sense that we can find the quantity it precedes under specified conditions.