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, etc.in Geometry, Algebra, and Calculus are
employed to solve complex problems in Mathematics.
