Friday, October 25, 2019

Methods to Prove Theorems in Mathematics

In Mathematics, a theorem is a logical consequence of the axioms. Theorems are an important part of Mathematics. Theorems have been proven on the basis of generally accepted statements, previously proven theorems. A theorem is a statement which is proven to be true. The proven statement which is used to prove another statement is called “Lemma”. A proof is a logical statement that tries to show the given statement is true. Generally, a theorem can be proved using three different ways. They are:

1.      Direct Proof
2.      Proof by contradiction
3.      Proof by Induction

Now, let us discuss these three different types of proofs below.

1.      Direct Proof

Direct Proof is the method of showing the truth or falsehood of the given statement using the straightforward facts, existing theorems and lemmas, and axioms without making further assumptions. Bayes Theorem is an example of direct proof where it proves the statement directly by using the existing theorem called total probability theorem.

2.      Proof by Contradiction

The term “ Contradiction” is used to describe the statement that involves opposing ideas. In Mathematics and logic, proof by contradiction is a method to prove the given statement by assuming the statement is false. We should show its falsity until the result of the assumption is a contraction. For example, we want to prove that the number √2 is a rational number. To prove this statement, we can use a proof by contradiction method by assuming “√2 is an irrational number,'' which is a false statement.

3.      Proof by Induction

A proof by induction is similar to an ordinary proof. But this technique employs a neat trick that allows you first to prove a statement about an arbitrary number and should determine the value greater than that. It means that proof by induction method needs two cases to be proved. The first case is termed as "base case" and the second case is termed as "induction step". The base case demonstrates the statement where its property holds for the number 0. The induction step proves that, if the characteristic of the statement exists for one natural number, say n, then its hold for the following natural number, say n+1. The base case need not begin with zero. 

Similarly, there are many theorems such as Pythagoras theorem, Stokes theorem, Green’s theorem, Geometry, Algebra, and Calculus are employed to solve complex problems in Mathematics.

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