Home > Well-founded relation

Note that this differs from the definition of the well-order relation, where the relation is required to be totally ordered and thus requires R- least elements instead of merely R-minimal ones.

A set equipped with a well-founded relation is sometimes said to be a well-founded set. A well-founded set is a partially ordered set in which every non-empty subset has a minimal element. Equivalently, assuming some choice, it is a partially ordered set which contains no infinite descending chains. If the order is a total order then the set is called a well-ordered set.

One reason that well-founded sets are interesting is because a version of transfinite induction can be used on them: if (X, <=) is a well-founded set and P(x) is some property of elements of X and you want to show that P(x) holds for all elements of X, you can proceed as follows:

1. show that P(x) is true for all minimal elements of X
2. show that, if x is an element of X and P(y) is true for all y <= x with y ≠ x, then P(x) must also be true.

Examples of well-founded sets which are not totally ordered are:

• the positive integers {1, 2, 3, ...}, with the order defined by a <= b iff a dividesNumber theory In mathematics, a divisor of an integer n also called a factor of n is an integer which evenly divides n without leaving a remainder. For example, 7 is a divisor of 42 because 42/7 6. We also say 42 is divisible by 7 or 42 is a multiple of 7 b.
• the set of all finite stringGenerally, string is a thin piece of fiber which is used to tie, bind, or hang other objects. String can be made from a variety of fibres. You can get different kinds, twine for example. The term has more specific meanings, within certain academic discipls over a fixed alphabet, with the order defined by s <= t iff s is a substring of t
• the set N × N of pairsIn mathematics, the Cartesian product (or direct product X × Y of two sets X and Y is the set of all ordered pairs whose first component is a member of X and whose second component is a member of Y''. This concept is named after Rene Descartes. X × Y { x of natural numberNatural number can mean either a positive integer ( 1, 2, 3, 4,. or a non-negative integer ( 0, 1, 2, 3, 4,. Natural numbers have two main purposes: they can be used for counting ("there are 3 apples on the table"), or they can be used for ordering ("thiss, ordered by (n₁, n₂) <= (m₁, m₂) iff n₁ ≤ m₁ and n₂ ≤ m₂.
• the set of all regular expressions over a fixed alphabet, with the order defined by s <= t iff s is a subexpression of t
• any set whose elements are sets, with the order defined by A <= B iff A is an element of B.

If (X, ≤) is a well-founded set and x is an element of X, then the descending chains starting at x are all finite, but this does not mean that their lengths are necessarily bounded. Consider the following example: Let X be the union of the positive integers and a new element ω, which is bigger than any integer. Then X is a well-founded set, but there are descending chains starting at ω of arbitrary great (finite) length; the chain ω, n-1, n-2, ..., 2, 1 has length n for any n.

A familiar example of a well-founded relation is the ordinary < relation on the set of natural numbers N. Every non-empty subset of the natural numbers contains a smallest element. This is known as the well-ordering principle.

Well-foundedness is interesting because the powerful technique of induction can be used to prove things about members of well-founded sets. For the example of the natural numbers above, this technique is called mathematical induction. When the well-founded set is the set of all ordinal numbers, and the well-founded relation is set membership, 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. See the articles under those heads for more details.