Mathematical Induction

Mathematical Induction is a method of supporting a mathematical statement. The method is based on producing evidence for the purpose of verification of the statement. The presented evidence serves to verify the mathematical statement. The correctness of the mathematical statement is tested by the verification.

The mathematical statement is verified. The mathematical statement stands true if use of scientific verification justifies the statement.

At first, the mathematical statement is verified by using a limited form of evidence. Then we go onward for further  verification. Onward verification also must justify the meaningfulness of the mathematical statement.

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Principle of Mathematical Induction:

Let P(n) be a statement involving the natural number n such that 

1. P(1) is true and

2. P(k+1) is true, whenever P(k) is true

then P(n) is true for all n € N.

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First the element produced as evidence is the natural number '1'. '1' is used in the mathematical statement. If the use of '1' leads to a successful test, we move forward. Verification of the statement using '1' is a partial proof of correctness of the mathematical statement.

The other evidence  can be considered as compound evidence based on two terms ( (k), (k+1) ). From the  first term (  (k) ), using P(k) foundation of further verification is laid.  The first term ( (k) ) is applied as P(k) to the mathematical statement. A result is formed.

Now, the second term ( (k+1) ) is applied to the mathematical statement as P(k+1). The result drawn by application of P(k) is lent to the mathematical statement formed by application of P(k+1). If the  mathematical statement formed by the application of P(k+1), using P(k),  is correct, then the original mathematical statement is correct.

We say that the original mathematical statement is true for all values.

IMITATING MATHEMATICAL INDUCTION:

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Statement: Hari attended the school today -> ( Suppose it to be P(n) )

Proof:

1 ) Hari was seen in school today by Jim. ( Verified by limited evidence similar to testing of mathematical statement by natural number '1') -> ( Similar to P(1) . Jim is a single person, similar to single value of natural number '1' )

Note1: The above statement can be supposed to be true.This is similar to verification of the   mathematical statement for '1'.

2 ) Hari attended his roll call today.  -> ( Similar to P(k) )

Note2: The original statement mentioned in the problem can be supposed to be true for P(k).  The original  statement is: Hari attended the school today. In a way this original statement is served by the statement: Hari attended his roll call today.

Hari's presence was found recorded in  the attendance register today.->  ( Similar to P(k+1) )

Note3: The statement analogous to P(k) was: Hari attended his roll call today. The statement for P(k+1)- Hari's presence was recorded in the attendance register today- is well served by the analogous version of the  statement for P(k) just presented.

Note: Had Hari not attended his roll call today, meaning  ( P(k) = ×), his presence would not have found recording in the attendance register today (P(k+1) = ×).

Because Hari attended his roll call today and hence P(k) has been committed and it is therefore true. 

Hari's presence has been recorded in the attendance register today and hence P(k+1) has been committed, and it is therefore true.

As Hari responds to his roll call today (P(k) is true), his attendance recorded in the attendance register is very truthful (P(k+1)) is true). So, here we have seen that if P(k) is true, P(k+1) is true. So, we say that Hari attended the school today,  and now everyone agrees with this. 

That's it.