Sets

In mathematics, a set can be thought of as any collection of distinct things considered as a whole. Though a simple idea, it is nevertheless one of the most important and fundamental concepts in modern mathematics, and the study of the structure of possible sets, set theory, is quite rich. more...

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Set theory, having only been invented at the end of the 19th century, is now a ubiquitous part of mathematics education, being introduced as early as primary school. It is the language in which modern mathematics is described. Set theory can be viewed as the foundation upon which nearly all of mathematics can be built and the source from which nearly all 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

A set is a collection of objects considered as a whole. The objects of a set are called elements or 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, A, B, C, etc. Two sets A and B are said to be equal, written A = B, if they have the same members.

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.

Describing sets

Not all sets have precise descriptions of any sort; they may simply be arbitrary collections, with no expressible "rule" saying what elements are in or out.

Some sets may be described in words, for example:

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 also be defined by explicitly listing its elements between braces (sometimes called curly brackets or curly braces), for example:

C = {4, 2, 1, 3}

D = {red, white, blue}

Two different descriptions may define the same set. For example, for the sets defined above, A and C are identical, since they have precisely the same members. The shorthand A = C is used to express this equality. Similarly, for the sets defined above, B = D.

Set identity does not depend on the order in which the elements are listed, nor on whether there are repetitions in the list. For example, {6, 11} = {11, 6} = {11, 11, 6, 11}.

For sets with many elements, an abbreviated list is sometimes used. For example, the first one thousand positive whole numbers can be described using the symbolic shorthand:

Read more at Wikipedia.org