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Induction in mathematics Definitions Applications Examples Induction Mathematical induction is a method of mathematical proof typically used to establish a given statement for all natural numbers. It is a form of direct proof, and it is done in two steps. The first step, known as the base case, is to prove the given statement for the first natural number. The second step, known as the inductive step, is to prove that the given statement for any one natural number implies the given statement for the next natural number. From these two steps, mathematical induction is the rule from which we infer that the given statement is established for all natural numbers. Induction (continued) 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. (continued) Induction Although its namesake may suggest otherwise, mathematical induction should not be misconstrued as a form of inductive reasoning (also see Problem of induction). Mathematical induction is an inference rule used in proofs. In mathematics, proofs including those using mathematical induction are examples of deductive reasoning and inductive reasoning is excluded from proofs. Induction (continued) (continued) Induction Induction (continued) (continued) Induction References and bibliography • Franklin, J.; A. Daoud (2011). Proof in Mathematics: An Introduction. Sydney: Kew Books. ISBN 0-646-54509-4. (Ch. 8.) • Hazewinkel, Michiel, ed. (2001), "Mathematical induction", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 • Knuth, Donald E. (1997). The Art of Computer Programming, Volume 1: Fundamental Algorithms (3rd ed.). Addison-Wesley. ISBN 0-201-89683-4. (Section 1.2.1: Mathematical Induction, pp. 11–21.) • Kolmogorov, Andrey N.; Sergei V. Fomin (1975). Introductory Real Analysis. Silverman, R. A. (trans., ed.). New York: Dover. ISBN 0-48661226-0. (Section 3.8: Transfinite induction, pp. 28–29.)
