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Sets

In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. Although this appears to be a simple idea, sets are one of the most important and fundamental concepts in modern mathematics. more...

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The study of the structure of possible sets, set theory, is rich and ongoing.

Having only been invented at the end of the 19th century, set theory is now a ubiquitous part of mathematics education, being introduced from primary school in many countries. Set theory can be viewed as the foundation upon which nearly all of mathematics can be derived. This article gives a brief and basic introduction to what mathematicians call "intuitive" or "naive" set theory; for a more detailed account see naive set theory. For a rigorous modern axiomatic treatment of sets, see axiomatic set theory.

Definition

At the beginning of his work BeitrÃ¤ge zur BegrÃ¼ndung der transfiniten Mengenlehre, Georg Cantor, the principal creator of set theory, made the following definition of a set:

The objects of a set are also called its members. The elements of a set can be anything: numbers, people, letters of the alphabet, other sets, and so on. Sets are conventionally denoted with capital letters, for instance A, B and C. Two sets A and B are said to be equal if they have the same members; this is written A = B.

A set, unlike a multiset, cannot contain two or more identical elements. All set operations preserve the property that each element in the set is unique. Similarly, the order in which the elements of a set are listed is irrelevant, unlike a sequence or tuple.

History

Applications

Set theory is seen as the foundation upon which virtually all of mathematics can be derived from. For example, structures in abstract algebra, such as groups, fields and rings, are sets closed under one or more operations.

Description

Not all sets have precise descriptions; they may be arbitrary collections, with no expressible inclusion criteria.

Some sets may be described in words:

A is the set whose members are the first four positive whole numbers.

B is the set whose members are the colors of the French flag.

By convention, a set can be defined by explicitly listing its elements between braces (sometimes called curly brackets or curly braces):

C = {4, 2, 1, 3}

D = {red, white, blue}

Two different descriptions may define the same set. Using the above examples, A and C are identical, since they have precisely the same members. The shorthand A = C is used to express this equality.