PIGEON HOLE PRINCIPLE AND MATHEMATICAL INDUCTION CONTRACTION

This paper tries to find the deficiencies an non- Universal validation of Pigeonhole Principle as well as established peano's axiom as principle of Mathematical Induction. There is emergent need to search the applicability of Real number systemin both of thecases. In the case of fractional particle which come out during reactions of different significant or non - significant needs based study for the industrial production process too may not be estimated by Pigeon hole principle or Principle of Mathematical Induction. Some other simulation process and treatments may not be limited to the scope of the set of Natural Number system that starts with unity and ends at infinity but what about Rational and irrational digits means a set of Real number system. Micro particles and their applicability and setting them to an extent level of assumption can make any Research task easy and significant for the further outcomes forecasting. Principle of Mathematical induction has been an important tool in proving any propositions or statements. Researchers have contributed more by saying that Pigeon  hole principle may be proved with the help of principle of Mathematical induction and vice versa but none of these two established principlespeak about  the Rational and irrational  digit taking case and analysis at these contexts, the enlistment rule is just an extraordinary instance of the categorize rule, and numerous combinatorial checking contentions lessen to the categorize rule and these two may have some lacuna in managing a few realities and structures of Analysis. Regardless of whether one can get rid of the feeble categorize rule, by demonstrating the presence of endlessly numerous primes inside I∆O.