This article is about 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

Set theory.

An example of a venn diagram

In mathematics, a set is a collection of distinct objects,

considered as an object in its own right. For example, the

numbers 2, 4, and 6 are distinct objects when considered

separately, but when they are considered collectively they form

a single set of size three, written {2,4,6}. Sets are one of the

most fundamental concepts in mathematics. Developed at the

end of the 19th century, set theory is now a ubiquitous part of

mathematics, and can be used as a foundation from which

nearly all of mathematics can be derived. In mathematics

education, elementary topics such as Venn diagrams are taught

at a young age, while more advanced concepts are taught as part

of a university degree. The term itself was coined by Bolzano in

his work The Paradoxes of the Infinite.

Definition

A set is a well defined collection of distinct objects. The objects

that make up a set (also known as the elements or members of a

set) can be anything: numbers, people, letters of the alphabet,

other sets, and so on. Georg Cantor, the founder of set theory,

gave the following definition of a set at the beginning of his

Beiträge zur Begründung der transfiniten Mengenlehre:[1]

“

A set is a gathering together into a whole of definite,

distinct objects of our perception [Anschauung] or of our

thought—which are called elements of the set.

”

Sets are conventionally denoted with capital letters. Sets A and

B are equal if and only if they have precisely the same elements.

[2]

As discussed below, the definition given above turned out to be

inadequate for formal mathematics; instead, the notion of a "set"

is taken as an undefined primitive in axiomatic set theory, and

its properties are defined by the Zermelo–Fraenkel axioms. The

most basic properties are that a set "has" elements, and that two

sets are equal (one and the same) if and only if every element of

one is an element of the other.

Describing sets

There are two ways of describing, or specifying the members of,

a set. One way is by intensional definition, using a rule or

semantic description:

A is the set whose members are the first four positive integers.

B is the set of colors of the French flag.

The second way is by extension – that is, listing each member of

the set. An extensional definition is denoted by enclosing the

list of members in curly brackets:

C = {4, 2, 1, 3}

D = {blue, white, red}.

Every element of a set must be unique; no two members may be

identical. (A multiset is a generalized concept of a set that

relaxes this criterion.) All set operations preserve this property.

The order in which the elements of a set or multiset are listed is

irrelevant (unlike for a sequence or tuple). Combining these two

ideas into an example

{6, 11} = {11, 6} = {11, 6, 6, 11}

because the extensional specification means merely that each of

the elements listed is a member of the set.

For sets with many elements, the enumeration of members can

be abbreviated. For instance, the set of the first thousand

positive integers may be specified extensionally as:

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

where the ellipsis ("...") indicates that the list continues in the

obvious way. Ellipses may also be used where sets have

infinitely many members. Thus the set of positive even numbers

can be written as {2, 4, 6, 8, ... }.

The notation with braces may also be used in an intensional

specification of a set. In this usage, the braces have the meaning

"the set of all ...". So, E = {playing card suits} is the set whose

four members are

,

,

, and

. A more general form of this is set-builder

notation, through which, for instance, the set F of the twenty

smallest integers that are four less than perfect squares can be

denoted:

F = {n2 − 4 : n is an integer; and 0 ≤ n ≤ 19}.

In this notation, the colon (":") means "such that", and the

description can be interpreted as "F is the set of all numbers of

the form n2 − 4, such that n is a whole number in the range from

0 to 19 inclusive." Sometimes the vertical bar ("|") is used

instead of the colon.

One often has the choice of specifying a set intensionally or

extensionally. In the examples above, for instance, A = C and B

= D.

Membership

Main article: Element (mathematics)

The key relation between sets is membership – when one set is

an element of another. If a is a member of B, this is denoted a ∈

B, while if c is not a member of B then c ∉ B. For example, with

respect to the sets A = {1,2,3,4}, B = {blue, white, red}, and F =

{n2 − 4 : n is an integer; and 0 ≤ n ≤ 19} defined above,

4 ∈ A and 12 ∈ F; but

9 ∉ F and green ∉ B.

Subsets

Main article: Subset

If every member of set A is also a member of set B, then A is

said to be a subset of B, written A ⊆ B (also pronounced A is

contained in B). Equivalently, we can write B ⊇ A, 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 ⊊ B (A is a proper subset of B) or B ⊋ A (B

is a proper superset of A).

Note that the expressions A ⊂ B and B ⊃ A are used differently by

different authors; some authors use them to mean the same as A

⊆ B (respectively B ⊇ A), whereas other use them to mean the

same as A ⊊ B (respectively B ⊋ A).

A is a subset of B

Example:

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

{1, 3} ⊆ {1, 2, 3, 4}.

{1, 2, 3, 4} ⊆ {1, 2, 3, 4}.

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

of itself:

∅ ⊆ A.

A ⊆ A.

An obvious but useful identity, which can often be used to show

that two seemingly different sets are equal:

A = B if and only if A ⊆ B and B ⊆ A.

A partition of a set S is a set of nonempty subsets of S such that

every element x in S is in exactly one of these subsets.

Power sets

Main article: Power set

The power set of a set S is the set of all subsets of S, including S

itself and the empty set. For example, the power set of the set {1,

2, 3} is {{1, 2, 3}, {1, 2}, {1, 3}, {2, 3}, {1}, {2}, {3}, ∅}. The

power set of a set S is usually written as P(S).

The power set of a finite set with n elements has 2n elements.

This relationship is one of the reasons for the terminology

power set[citation needed]. For example, the set {1, 2, 3}

contains three elements, and the power set shown above

contains 23 = 8 elements.

The power set of an infinite (either countable or uncountable)

set is always uncountable. Moreover, the power set of a set is

always strictly "bigger" than the original set in the sense that

there is no way to pair the elements of a set S with the elements

of its power set P(S) such that every element of S set is paired

with exactly one element of P(S), and every element of P(S) is

paired with exactly one element of S. (There is never a bijection

from S onto P(S).)

Every partition of a set S is a subset of the powerset of S.

Cardinality

Main article: Cardinality

The cardinality | S | of a set S is "the number of members of S."

For example, if B = {blue, white, red}, | B | = 3.

There is a unique set with no members and zero cardinality,

which is called the empty set (or the null set) and is denoted by

the symbol ∅ (other notations are used; see empty set). For

example, the set of all three-sided squares has zero members

and thus is the empty set. Though it may seem trivial, the empty

set, like the number zero, is important in mathematics; indeed,

the existence of this set is one of the fundamental concepts of

axiomatic set theory.

Some sets have infinite cardinality. The set N of natural

numbers, for instance, is infinite. Some infinite cardinalities are

greater than others. For instance, the set of real numbers has

greater cardinality than the set of natural numbers. However, it

can be shown that the cardinality of (which is to say, the number

of points on) a straight line is the same as the cardinality of any

segment of that line, of the entire plane, and indeed of any

finite-dimensional Euclidean space.

Special sets

There are some sets that hold great mathematical importance

and are referred to with such regularity that they have acquired

special names and notational conventions to identify them. One

of these is the empty set, denoted {} or ∅. Another is the unit set

{x}, which contains exactly one element, namely x.[2] Many of

these sets are represented using blackboard bold or bold

typeface. Special sets of numbers include:

P or ℙ, denoting the set of all primes: P = {2, 3, 5, 7, 11, 13,

17, ...}.

N or ℕ, denoting the set of all natural numbers: N = {1, 2,

3, . . .} (sometimes defined containing 0).

Z or ℤ, denoting the set of all integers (whether positive,

negative or zero): Z = {..., −2, −1, 0, 1, 2, ...}.

Q or ℚ, denoting the set of all rational numbers (that is, the set

of all proper and improper fractions): Q = {a/b : a, b ∈ Z, b ≠

0}. For example, 1/4 ∈ Q and 11/6 ∈ Q. All integers are in this

set since every integer a can be expressed as the fraction a/1

(Z ⊊ Q).

R or ℝ, denoting the set of all real numbers. This set includes

all rational numbers, together with all irrational numbers (that

is, numbers that cannot be rewritten as fractions, such as √2, as

well as transcendental numbers such as π, e and numbers that

cannot be defined).

C or ℂ, denoting the set of all complex numbers: C = {a + bi :

a, b ∈ R}. For example, 1 + 2i ∈ C.

H or ℍ, denoting the set of all quaternions: H = {a + bi + cj +

dk : a, b, c, d ∈ R}. For example, 1 + i + 2j − k ∈ H.

Positive and negative sets are denoted by a superscript - or +,

for example: ℚ+ represents the set of positive rational numbers.

Each of the above sets of numbers has an infinite number of

elements, and each can be considered to be a proper subset of

the sets listed below it. The primes are used less frequently than

the others outside of number theory and related fields.

Basic operations

There are several fundamental operations for constructing new

sets from given sets.

Unions

The union of A and B, denoted A ∪ B

Main article: Union (set theory)

Two sets can be "added" together. The union of A and B,

denoted by A ∪ B, is the set of all things that are members of

either A or B.

Examples:

{1, 2} ∪ {1, 2} = {1, 2}.

{1, 2} ∪ {2, 3} = {1, 2, 3}.

Some basic properties of unions:

A ∪ B = B ∪ A.

A ∪ (B ∪ C) = (A ∪ B) ∪ C.

A ⊆ (A ∪ B).

A ∪ A = A.

A ∪ ∅ = A.

A ⊆ B if and only if A ∪ B = B.

Intersections

Main article: Intersection (set theory)

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 that are members of

both A and B. If A ∩ B = ∅, then A and B are said to be disjoint.

The intersection of A and B, denoted A ∩ B.

Examples:

{1, 2} ∩ {1, 2} = {1, 2}.

{1, 2} ∩ {2, 3} = {2}.

Some basic properties of intersections:

A ∩ B = B ∩ A.

A ∩ (B ∩ C) = (A ∩ B) ∩ C.

A ∩ B ⊆ A.

A ∩ A = A.

A ∩ ∅ = ∅.

A ⊆ B if and only if A ∩ B = A.

Complements

The relative complement of B in A

The complement of A in U

The symmetric difference of A and B

Main article: Complement (set theory)

Two sets can also be "subtracted". The relative complement of B

in A (also called the set-theoretic difference of A and B),

denoted by A \ B (or A − B), is the set of all elements that are

members of A but not members of B. Note that it is valid to

"subtract" members of a set that are not in the set, such as

removing the element green from the set {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′.

Examples:

{1, 2} \ {1, 2} = ∅.

{1, 2, 3, 4} \ {1, 3} = {2, 4}.

If U is the set of integers, E is the set of even integers, and

O is the set of odd integers, then U \ E = E′ = O.

Some basic properties of complements:

A \ B ≠ B \ A for A ≠ B.

A ∪ A′ = U.

A ∩ A′ = ∅.

(A′)′ = A.

A \ A = ∅.

U′ = ∅ and ∅′ = U.

A \ B = A ∩ B′.

An extension of the complement is the symmetric difference,

defined for sets A, B as

For example, the symmetric difference of {7,8,9,10} and

{9,10,11,12} is the set {7,8,11,12}.

Cartesian product

Main article: Cartesian product

A new set can be constructed by associating every element of

one set with every element of another set. The Cartesian

product of two sets A and B, denoted by A × B is the set of all

ordered pairs (a, b) such that a is a member of A and b is a

member of B.

Examples:

{1, 2} × {red, white} = {(1, red), (1, white), (2, red), (2,

white)}.

{1, 2} × {red, white, green} = {(1, red), (1, white), (1,

green), (2, red), (2, white), (2, green) }.

{1, 2} × {1, 2} = {(1, 1), (1, 2), (2, 1), (2, 2)}.

Some basic properties of cartesian products:

A × ∅ = ∅.

A × (B ∪ C) = (A × B) ∪ (A × C).

(A ∪ B) × C = (A × C) ∪ (B × C).

Let A and B be finite sets. Then

| A × B | = | B × A | = | A | × | B |.

Applications

Set theory is seen as the foundation from which virtually all of

mathematics can be derived. For example, structures in abstract

algebra, such as groups, fields and rings, are sets closed under

one or more operations.

One of the main applications of naive set theory is constructing

relations. A relation from a domain A to a codomain B is a

subset of the Cartesian product A × B. Given this concept, we

are quick to see that the set F of all ordered pairs (x, x2), where

x is real, is quite familiar. It has a domain set R and a codomain

set that is also R, because the set of all squares is subset of the

set of all reals. If placed in functional notation, this relation

becomes f(x) = x2. The reason these two are equivalent is for

any given value, y that the function is defined for, its

corresponding ordered pair, (y, y2) is a member of the set F.

Axiomatic set theory

Main article: Axiomatic set theory

Although initially naive set theory, which defines a set merely

as any well-defined collection, was well accepted, it soon ran

into several obstacles. It was found that this definition spawned

several paradoxes, most notably:

Russell's paradox—It shows that the "set of all sets that do not

contain themselves," i.e. the "set" { x : x is a set and x ∉ x }

does not exist.

Cantor's paradox—It shows that "the set of all sets" cannot

exist.

The reason is that the phrase well-defined is not very well

defined. It was important to free set theory of these paradoxes

because nearly all of mathematics was being redefined in terms

of set theory. In an attempt to avoid these paradoxes, set theory

was axiomatized based on first-order logic, and thus axiomatic

set theory was born.

For most purposes however, naive set theory is still useful.

Principle of inclusion and exclusion

Main article: Inclusion-exclusion principle

This principle gives us the cardinality of the union of sets.