Methods of Mathematical Proof

Introduction to methods of mathematical proof:

Proof is used to show the truth of a statement. We can show the mathematical statements in different methods. Direct proof, indirect proof or proof by contradiction, proof by counter example, geometrical proof technique, and proof by construction are the methods of mathematical proofs. Let us see methods of mathematical proof in this article.
Methods of Mathematical Proof:

Mathematical proof:

Meaning of the statement proof is verification, confirmation, etc. Proof is applied to prove the truth of a statement

Methods of mathematical proof:

Direct proof
Indirect proof or proof by contradiction
Proof by counter example
Geometrical proof technique
Proof by construction

Concept of direct proof:

Let us consider we have to prove that P`rArr` Q. Consider P is true. By the step by step explanation, we find that Q is true. This method of proving Q is true from that P is true is called as direct proof method.

Concept of indirect proof:

Let us consider we have to prove that P `rArr`Q. Initially assume P is true and Q is not true. By the step by step explanation, we appear at an opposite to our assumption. This method of proving is called as indirect proof method.

Concept of proof by counter example:

Assume ‘A’ and ‘B’ are two statements. Let us consider that we want to know whether A B. if we able to find an example wherever A is true but B is false, then we conclude that A B. The example which has established it is called a counter example.

Geometrical proof technique:

Several algebraic problems can be solved by geometrical techniques.

Concept of proof by construction:

When we explain some of the problems in geometry, we perform some middle constructions like placing lines to the figure in order to find the solution. Construction is a brilliant method in many geometrical problems. But construction should be finished at appropriate places. Thus proof by construction needs fore-sightedness on the part of the problem solver.
Problems for Methods of Mathematical Proof:

Problem 1:

Prove that if 2 sides of a triangle are same, then the angles opposite to the triangle are equal.

Solution:

Let draw a triangle ABC. Assume that the length of AB is equal to the length of AC. We want to show that `angleB` = `angleC` .

 

methods of mathematical proof

In above diagram D is a mid point of the line BC. And the point A is joined with D. now we have two triangles ABD and ACD. In these two triangles the length of BD is equal to the length of DC, the length of AB is equal to the length of AC and AD is a common to two triangles. Therefore the triangles are congruent then the corresponding angles are equal so`angleB` = `angleC`.

Problem 2:

If x = 2 and y = 1 then prove (x-y)2 =1

Solution:

Given

x = 2

y = 1

(x - y)2 = x2 + y2 - 2xy

(x - y)2 = (2)2 + (1)2 - 2(2)(1)

(x - y)2 = 4 + 1 - 4

(x - y)2 = 1