Recursive definition  

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Train wreck at Montparnasse (October 22, 1895) by Studio Lévy and Sons.
Train wreck at Montparnasse (October 22, 1895) by Studio Lévy and Sons.

A recursive definition (or inductive definition) in mathematical logic and computer science is used to define the elements in a set in terms of other elements in the set (Aczel 1978:740ff).

A recursive definition of a function defines values of the functions for some inputs in terms of the values of the same function for other inputs. For example, the factorial function n! is defined by the rules

0! = 1.
(n+1)! = (n+1)·n!.

This definition is valid for all n, because the recursion eventually reaches the base case of 0. The definition may also be thought of as giving a procedure describing how to construct the function n!, starting from n = 0 and proceeding onwards with n = 1, n = 2, n = 3 etc..

The recursion theorem states that such a definition indeed defines a function. The proof uses mathematical induction.

An inductive definition of a set describes the elements in a set in terms of other elements in the set. For example, one definition of the set N of natural numbers is:

  1. 1 is in N.
  2. If an element n is in N then n+1 is in N.
  3. N is the intersection of all sets satisfying (1) and (2).

There are many sets that satisfy (1) and (2) - for example, the set {1, 1.649, 2, 2.649, 3, 3.649, ...} satisfies the definition. However, condition (3) specifies the set of natural numbers by removing the sets with extraneous members.

Properties of recursively defined functions and sets can often be proved by an induction principle that follows the recursive definition. For example, the definition of the natural numbers presented here directly implies the principle of mathematical induction for natural numbers: if a property holds of the natural number 0, and the property holds of n+1 whenever it holds of n, then the property holds of all natural numbers (Aczel 1978:742).

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