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Course overview (in Hindi)

Lesson 1 of 5 ‚ÄĘ 25 upvotes ‚ÄĘ 4:32 mins

Jagat Chaudhary

Mathematical induction¬†is a mathematical proof technique. It is essentially used to prove that a property¬†P(n) holds for every¬†natural number¬†n, i.e. for¬†n¬†=¬†0,¬†1,¬†2,¬†3,¬†and so on. Metaphors can be informally used to understand the concept of mathematical induction, such as the metaphor of falling dominoes or climbing a ladder: Mathematical induction proves that we can climb as high as we like on a ladder, by proving that we can climb onto the bottom rung (the¬†basis) and that from each rung we can climb up to the next one (the¬†step). ‚ÄĒ‚ÄČConcrete Mathematics The method of induction requires two cases to be proved. The first case, called the¬†base case¬†(or, sometimes, the¬†basis), proves that the property holds for the number 0. The second case, called the¬†induction step, proves that, if the property holds for one natural number¬†n, then it holds for the next natural number¬†n¬†+¬†1. These two steps establish the property¬†P(n) for every natural number¬†n¬†=¬†0,¬†1,¬†2,¬†3,¬†... The base step need not begin with zero. Often it begins with the number one, and it can begin with any natural number, establishing the truth of the property for all natural numbers greater than or equal to the starting number. The method can be extended to prove statements about more general¬†well-founded¬†structures, such as¬†trees; this generalization, known as¬†structural induction, is used in¬†mathematical logic¬†and¬†computer science. Mathematical induction in this extended sense is closely related to¬†recursion. Mathematical induction, in some form, is the foundation of all¬†correctness proofs for computer programs