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Contents
1 Induction and recursion
2 Examples
3 Other properties
4 Reflexivity

Induction and recursion
On par with induction, well-founded relations also support construction of objects by transfinite recursion. Let (X, R) be a set-like well-founded relation and F a function that assigns an object F(x, g) to each pair of an element x ∈ X and a function g on the initial segment {y: y R x} of X. Then there is a unique function G such that for every x ∈ X,

As an example, consider the well-founded relation (N, S), where N is the set of all natural numbers, and S is the graph of the successor function x → x + 1. Then induction on S is the usual mathematical induction, and recursion on S gives primitive recursion.
If we consider the order relation (N, <), we obtain complete induction, and course-of-values recursion. The statement that (N, <) is well-founded is also known as the well-ordering principle.

There are other interesting special cases of well-founded induction. When the well-founded relation is the usual ordering on the class of all ordinal numbers, the technique is called transfinite induction. When the well-founded set is a set of recursively-defined data structures, the technique is called structural induction.
When the well-founded relation is set membership on the universal class, the technique is known as ∈-induction. See those articles for more details.

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