Solving Mathematical Problem

We , means Human and Computer both can Solve only finite problems [ Problems which can be solved by finite number of Computation(s) ] .

There are some problems which can't be solved by Any Human / Computer in Countable time [ They are called Undecidable Problem ] .

So we will solve problems Using following Steps : ->

1. Determine whether Problem is Decidable .

2. If Decidable then see whether it is finite or not.

3. If finite then solve it in finite steps else follow :-

              a) Induction       or 

              b) Contradiction  which one is Useful to reduce problem in finite Domain 

Induction:-

       

       Chose a property say P s.t. if it is valid for the whole set of problem then we can obviously say that the      

       problem is solved .  

      Find a Bijection f from problem set  to set of Natural Number N s.t Problem Set  = f(n)  

      where n ∈ N

      

       Basis: Prove it for a Small Cluster Of Problem Set .

       

       

       Induction Step : Prove that if for Problem set  f(1),f(2)...f(n) P holds then for f(n+1) also it holds.

       Now we can say that P holds for the Whole Problem Set . 

       

      That means Problem Solved.

Example :- Prove that f(n)=1+2+...n =n(n+1)/2 

                  Basis: n=1 it is true 

                  Induction Step : f(n)=n(n+1)/2 

                                          f(n+1)=f(n)+n+1= (n+1)(n+2)/2 i.e true

                                        

                 Proved.

Contradiction: Let the opposite is true . Then do the calculation with other Properties and  eventually you will  

                      find that a contradiction will arrive .

Example:- x=2 y=3 z=x+y then show z=5 

                let z!=5 

                x+y=2+3=5

                z=x+y

                5!=5 contradiction .