RBSE Class 11 Maths Notes Chapter 4 Principle of Mathematical Induction

These comprehensive RBSE Class 11 Maths Notes Chapter 4 Principle of Mathematical Induction will give a brief overview of all the concepts.

RBSE Class 11 Maths Chapter 4 Notes Principle of Mathematical Induction

Introduction
Numbers one, two, three, four, five are denoted by international symbols 1, 2, 3, 4, 5, .... From ancient time these are used in mathematical calculations.
In practical problems, numbers are used in natural form. So these numbers are called natural numbers. Set of natural numbers is denoted by N. .
On the basis of some results, we cannot draw general result. Thus, to obtain general result, a definite process is required. This definite process is called theory of Mathematical Induction.
With the help of general mathematical result a specific mathematical result is concluded. This is called as Mathematical Conclusion.

Statement
A line which means true or false is called statement.

For example :
New Delhi is the capital of India which means true. So, this is a statement.
Similarly, ‘Zero’ is a natural number. So, this is a statement.
There are mainly three types of statements :
(1) General Statement :
If sum of digit of any number is divisible by 3, then number is divisible by 3.

(2) Specific Statement:
210 is divisible by 3.
∵ Sum of digits of number 210 = 2 + 1+ 0 = 3
It is divisible by 3.
Hence, number 210 is also divisible by 3.

(3) Mathematical Statement:
The statement in which mathematical relation is expected is called mathematical statement. Mathematical statement is shown by P(n) where n is a natural number.
For Example :
3 + 5 = 8, A ∪ B = B ∪ A, are mathematical statements.

Successor and Contradiction
Successor : A number exactly after the natural number n i.e., n + 1 is called successor of n. i.e., if = n + 1
Contradiction : It is not necessary' that result obtained from mathematical conclusion is true because these results are inspired by some specific eases. As : by putting n = 1, 2, .............. 39 in n² + n + 41, we get prime number. But by putting n = 40 we do not get prime number.

First Principle of Mathematical Induction
According to this principle any statement P(n) is true for all natural numbers n. If
(i) P(1) is true i.e., given statement is true forn = 1 and
(ii) P(m) is true then P(m + 1) will be true.
If the statement is true for n = m then it is also true for n = m + 1. The first step of principle is simply a statement of fact. In this step,-we show that given statement is true for n = 1 but if given statement is true for n ≥ i in this step, we will show then that statement is true for n = i in this step, we will show then that statement is true for n = i in place of n = 1.
The next step of principle is called the induction step. Here we suppose that statement is true for n = m and prove that statement is also true for n = m + 1.
For example,
(i) P(n) = 3²ⁿ - 1, where n ∈ N, is divisible by 8.
(ii) P(n) = 2 + 4 + ... + 2n = n(n + 1), where n ∈ N.
Proof :
Let M be subset of set of natural numbers N and any statement P(m) is true for each m ∈ M
i.e., P(m) is true ∀ m ∈ M
Now, if
(i) P( 1) is true then 1 ∈ M
(ii) P(A) is true then P(A ) or P(A + 1) is true then
K ∈ M ⇒ k⁺ ∈ M[(k + 1) ∈ M]
Thus M = N [By Piano’s Axiom]
i.e., statement P(n), is true for all values of n. Induction theory is very powerful tool to prove theorems. To use mathematical induction theory, first we will show given statement is true for n = 1 and then prove it is true for any natural number A hence, it is also true for k + 1.

Second Principle of Mathematical Induction
Let {P(n): n ∈ N} is set of statements for each natural number such that
(i) P(1) is true, i.e., for n = 1, P(n) is true and
(ii) For each natural number k truth of statement P(m) (where m < k) represents truth of P(A). then statement P(«) is true for all natural numbers n.