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.

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.

As opposed to a multiset and a real-life collection, a set cannot contain multiple copies of an element.

Describing sets

Descriptions using words or lists

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}

Notice that 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}.

Descriptions using mathematical notation

For large sets (that is to say, sets in which there are many elements), it becomes highly impractical to explicitly write out the full list of contents. For example, E = {the first one thousand positive whole numbers} would, as a list, be as tedious to write as it would be to read. However, a mathematician would seldom describe E in words as above, preferring instead to use a symbolic shorthand:

E = {1, 2, 3, ..., 1000}

An abbreviated list can be used to describe a set such as E, where the elements can follow a pattern that is obvious to the reader. The full list is abbreviated using the ellipsis (...) symbol. When using this notation, care should also be taken to give enough elements to make the pattern clear. For example, the following set could, depending on the context, reasonably refer to either the first sixteen whole numbers or the first five powers of two:

X = {1, 2, ..., 16}

If, on the other hand, the characterizing property describes a less obvious pattern, then it is ill-advised to use an abbreviated list, which could serve to confuse the reader. For example, upon reading

F = {–4, –3, 0, ..., 357}

it is unclear that

F = {the first twenty numbers which are four less than a square number}.

In such circumstances, mathematicians describe the characterizing property of the set using mathematical notation. For example:

F = { – 4 : n is a whole number and 0 ≤ n ≤ 19}

In this description, the colon (:) means such that, and the mathematician interprets this description as

F is the set of numbers of the form – 4, such that n is a whole number in the range from 0 to 19 inclusive. (Sometimes the pipe notation | is used instead of the colon.)

An explicit list of the contents of F can be found by evaluating the expression – 4 for each value of n from 0 to 19.

Set membership

If something is or is not an element of a particular set then this is symbolised by and respectively. So, for example, with respect to the sets defined above:

• and (since 285 = 17² − 4); but
• and .

Cardinality of a set

Each of the sets described above has a definite number of members; for example, the set A has four members, while the set B has three members.

A set can also have zero members. Such a set is called the empty set (or the null set) and is denoted by the symbol ø. For example, the set A of all three-sided squares has zero members, and thus A = ø. Like the number zero, though seemingly trivial, the empty set turns out to be quite important in mathematics.

A set can also have an infinite number of members; for example, the set of natural numbers is infinite.

For more information on infinity and the size of sets, see cardinality and cardinal number.

For more information on finite sets and counting them, see combinatorics and permutations and combinations.

Subsets

If every member of the set A is also a member of the set B, then A is said to be a subset of B, written , also pronounced A is contained in B. Equivalently, we can write , read as B is a superset of A, B includes A, or B contains A. The relationship between sets established by is called inclusion or containment.

If A is a subset of but not equal to B, then A is called a proper subset of B, written (A is a proper subset of B) or (B is proper superset of A). However, in some literature these symbols are read the same as and , so it's often preferred to use the more explicit symbols and for proper subsets and supersets.

A is a subset of B

Examples:

• The set of all men is a proper subset of the set of all people.

The empty set is a subset of every set and every set is a subset of itself:

Special sets

There are some sets which hold great mathematical importance and are referred to with such regularity that they have acquired special names to identify them. One of these is the empty set. Some special sets of numbers include:

• denotes the set of all natural numbers. That is to say, = {1, 2, 3, ...}, or sometimes = {0, 1, 2, 3, ...}.
• denotes the set of all integers (whether positive, negative or zero). So = {..., -2, -1, 0, 1, 2, ...}.
• denotes the set of all rational numbers (that is, the set of all proper and improper fractions). So, = { : a,b and b ≠ 0}. For example, and . All integers are in this set since every integer a can be expressed as the fraction .
• is the set of all real numbers. This set includes all rational numbers, together with all irrational numbers (that is, numbers which can't be rewritten as fractions, such as and √2).
• is the set of all complex numbers.

Each of these sets of numbers has infinite cardinality, and moreover .

Unions

There are several ways to construct new sets from existing ones. Two sets can be "added" together. The union of A and B, denoted by A U B, is the set of all things which are members of either A or B.

The union of A and B

Examples:

• {1, 2} U {red, white} = {1, 2, red, white}
• {1, 2, green} U {red, white, green} = {1, 2, red, white, green}
• {1, 2} U {1, 2} = {1, 2}

Some basic properties of unions:

• A U B   =   B U A
• A  is a subset of  A U B
• A U A   =  A
• A U ø   =  A

Intersections

A new set can also be constructed by determining which members two sets have "in common". The intersection of A and B, denoted by A ∩ B, is the set of all things which are members of both A and B. If A ∩ B  =  ø, then A and B are said to be disjoint.

The intersection of A and B

Examples:

• {1, 2} ∩ {red, white} = ø
• {1, 2, green} ∩ {red, white, green} = {green}
• {1, 2} ∩ {1, 2} = {1, 2}

Some basic properties of intersections:

• A ∩ B   =   B ∩ A
• A ∩ B  is a subset of  A
• A ∩ A   =   A
• A ∩ ø   =   ø

Complements

Two sets can also be "subtracted". The relative complement of A in B (also called the set theoretic difference of B and A), denoted by B − A, (or B \ A) is the set of all elements which are members of B, but not members of A. Note that it is valid to "subtract" members of a set that are not in the set, such as removing green from {1,2,3}; doing so has no effect.

In certain settings all sets under discussion are considered to be subsets of a given universal set U. In such cases, U − A, is called the absolute complement or simply complement of A, and is denoted by A′.

The relative complement
of A in B
The complement of A in U

Examples:

• {1, 2} − {red, white} = {1, 2}
• {1, 2, green} − {red, white, green} = {1, 2}
• {1, 2} − {1, 2} = ø
• If U is the set of integers, E is the set of even integers, and O is the set of odd integers, then the complement of E in U is O, or equivalently, E′ = O.

Some basic properties of complements:

• A U A′ = U
• A ∩ A′ = ø
• (A′ )′ = A
• A − A = ø
• A − B = A ∩ B′

For more information on the basic properties of sets, subsets, intersections, unions and complements, see algebra of sets. For a more general development of these ideas and others in set theory, see naive set theory.

• Alternative set theory
• Class (set theory)
• Family (mathematics)
• Mathematical structure
• Multiset