# Maximal Set

__Keywords__: Theory

In mathematics, specifically set theory, the term **maximal set** is often used to define specific sets in an elegant manner. In layman's terms, a maximal set describes a set that is "as large as possible". Without using this term, definitions of particular sets can get very cumbersome.

Specifically, suppose we want to describe a set *A* that satisfies some property *P*, but so that no proper subset of *A* should be included in the definition of such a set. Put another way, we want a set *A* satisfying the following two conditions:

- the set
*A*has property*P*; - if
*B*is another set with property*P*, and if*B*contains^{[1]}*A*, then*B*must be equal to*A*.

Then we simply say that *A* is a **maximal set** that satisfies property *P*.

Note that maximal sets need not be unique. That is, there can be multiple maximal sets.

As an example, we can define a Black chain to be a maximal set of connected Black intersections. We consider each chain as a single unit, and we don't call a three-stone connected subset of a five-stone chain a "chain".

Note also that *maximal* does not mean the same as *biggest* (in the sense of largest number of elements). For example, suppose there is a five-stone chain on the board as described above, and there is also a six-stone chain elsewhere on the board. Then both chains are maximal in the sense described here. The six-stone chain has a *larger* number of stones than the five-stone chain, but it does not *contain* it.

See also: Wikipedia pages for set theory and Maximal element.

[1] Mathematically speaking, "*B* contains *A*" means that every member of *A* is also a member of *B*; this includes the possibility that *A* and *B* are in fact equal.