A set is the mathematical model for a collection of different things;
a set contains
elements or
members, which can be mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets. The set with no element is the
empty set; a set with a single element is a singleton. A set may have a finite number of elements or be an
infinite set. Two sets are equal if they have precisely the same elements.
Sets are ubiquitous in modern mathematics. Indeed, set theory, more specifically Zermelo–Fraenkel set theory, has been the standard way to provide rigorous foundations for all branches of mathematics since the first half of the 20th century.
History
The concept of a set emerged in mathematics at the end of the 19th century.
The German word for set,
Menge, was coined by
Bernard Bolzano in his work
Paradoxes of the Infinite.
Georg Cantor, one of the founders of set theory, gave the following definition at the beginning of his
Beiträge zur Begründung der transfiniten Mengenlehre:
Bertrand Russell called a set a class:[Bertrand Russell (1903) The Principles of Mathematics, chapter VI: Classes]
Naive set theory
The foremost property of a set is that it can have elements, also called
members. Two sets are equal when they have the same elements. More precisely, sets
A and
B are equal if every element of
A is an element of
B, and every element of
B is an element of
A; this property is called the
extensionality of sets.
The simple concept of a set has proved enormously useful in mathematics, but paradoxes arise if no restrictions are placed on how sets can be constructed:

Russell's paradox shows that the "set of all sets that do not contain themselves", i.e., , cannot exist.

Cantor's paradox shows that "the set of all sets" cannot exist.
Naïve set theory defines a set as any welldefined collection of distinct elements, but problems arise from the vagueness of the term welldefined.
Axiomatic set theory
In subsequent efforts to resolve these paradoxes since the time of the original formulation of naïve set theory, the properties of sets have been defined by
axioms. Axiomatic set theory takes the concept of a set as a
primitive notion.
The purpose of the axioms is to provide a basic framework from which to deduce the truth or falsity of particular mathematical propositions (statements) about sets, using firstorder logic. According to Gödel's incompleteness theorems however, it is not possible to use firstorder logic to prove any such particular axiomatic set theory is free from paradox.
How sets are defined and set notation
Mathematical texts commonly denote sets by
capital letters in
Italic type, such as , , .
A set may also be called a
collection or
family, especially when its elements are themselves sets.
Roster notation
Roster or
enumeration notation defines a set by listing its elements between
, separated by commas:
 .
In a set, all that matters is whether each element is in it or not, so the ordering of the elements in roster notation is irrelevant (in contrast, in a sequence, a tuple, or a permutation of a set, the ordering of the terms matters). For example, and represent the same set.
For sets with many elements, especially those following an implicit pattern, the list of members can be abbreviated using an ellipsis ''.
For instance, the set of the first thousand positive integers may be specified in roster notation as
 .
Infinite sets in roster notation
An
infinite set is a set with an endless list of elements. To describe an infinite set in roster notation, an ellipsis is placed at the end of the list, or at both ends, to indicate that the list continues forever. For example, the set of
natural number is
and the set of all
is
Semantic definition
Another way to define a set is to use a rule to determine what the elements are:
 Let be the set whose members are the first four positive integer.
 Let be the set of colors of the French flag.
Such a definition is called a semantic description.
Setbuilder notation
Setbuilder notation specifies a set as a selection from a larger set, determined by a condition on the elements.
For example, a set can be defined as follows:
 $=\; \backslash \{n\; \backslash mid\; n\; \backslash text\{\; is\; an\; integer,\; and\; \}\; 0\; \backslash leq\; n\; \backslash leq\; 19\backslash \}.$
In this notation, the vertical bar "" means "such that", and the description can be interpreted as " is the set of all numbers such that is an integer in the range from 0 to 19 inclusive". Some authors use a colon ":" instead of the vertical bar.
Classifying methods of definition
Philosophy uses specific terms to classify types of definitions:

An intensional definition uses a rule to determine membership. Semantic definitions and definitions using setbuilder notation are examples.

An extensional definition describes a set by listing all its elements.
[ Such definitions are also called enumerative.
]

An ostensive definition is one that describes a set by giving examples of elements; a roster involving an ellipsis would be an example.
Membership
If is a set and is an element of , this is written in shorthand as , which can also be read as "
x belongs to
B", or "
x is in
B". The statement "
y is not an element of
B" is written as , which can also be read as "
y is not in
B".
For example, with respect to the sets , , and ,
 and ; and
 and .
The empty set
The
empty set (or
null set) is the unique set that has no members. It is denoted or
$\backslash emptyset$ or
or
(or ).
Singleton sets
A
singleton set is a set with exactly one element; such a set may also be called a
unit set.
[ Any such set can be written as , where x is the element.
The set and the element x mean different things; Halmos draws the analogy that a box containing a hat is not the same as the hat.
]
Subsets
If every element of set A is also in B, then A is described as being a subset of B, or contained in B, written A ⊆ B, or B ⊇ A. The latter notation may be read B contains A, B includes A, or B is a superset of A. The relationship between sets established by ⊆ is called inclusion or containment. Two sets are equal if they contain each other: A ⊆ B and B ⊆ A is equivalent to A = B.[
]
If A is a subset of B, but A is not equal to B, then A is called a proper subset of B. This can be written A ⊊ B. Likewise, B ⊋ A means B is a proper superset of A, i.e. B contains A, and is not equal to A.
A third pair of operators ⊂ and ⊃ are used differently by different authors: some authors use A ⊂ B and B ⊃ A to mean A is any subset of B (and not necessarily a proper subset),[ while others reserve A ⊂ B and B ⊃ A for cases where A is a proper subset of B.][
]
Examples:

The set of all humans is a proper subset of the set of all mammals.

⊂ .

⊆ .
The empty set is a subset of every set, and every set is a subset of itself:
Euler and Venn diagrams
An Euler diagram is a graphical representation of a collection of sets; each set is depicted as a planar region enclosed by a loop, with its elements inside. If is a subset of , then the region representing is completely inside the region representing . If two sets have no elements in common, the regions do not overlap.
A Venn diagram, in contrast, is a graphical representation of sets in which the loops divide the plane into zones such that for each way of selecting some of the sets (possibly all or none), there is a zone for the elements that belong to all the selected sets and none of the others. For example, if the sets are , , and , there should be a zone for the elements that are inside and and outside (even if such elements do not exist).
Special sets of numbers in mathematics
There are sets of such mathematical importance, to which mathematicians refer so frequently, that they have acquired special names and notational conventions to identify them.
Many of these important sets are represented in mathematical texts using bold (e.g. $\backslash bold\; Z$) or blackboard bold (e.g. $\backslash mathbb\; Z$) typeface.
These include

$\backslash bold\; N$ or $\backslash mathbb\; N$, the set of all : $\backslash bold\; N=\backslash \{0,1,2,3,...\backslash \}$ (often, authors exclude );
[
]

$\backslash bold\; Z$ or $\backslash mathbb\; Z$, the set of all (whether positive, negative or zero): $\backslash bold\; Z=\backslash \{...,2,1,0,1,2,3,...\backslash \}$;
[
]

$\backslash bold\; Q$ or $\backslash mathbb\; Q$, the set of all (that is, the set of all proper fraction and improper fractions): $\backslash bold\; Q=\backslash left\backslash \{\backslash frac\; \{a\}\{b\}\backslash mid\; a,b\backslash in\backslash bold\; Z,b\backslash ne0\backslash right\backslash \}$. For example, and ;
[
]

$\backslash bold\; R$ or $\backslash mathbb\; R$, the set of all real number, including all rational numbers and all irrational numbers (which include such as $\backslash sqrt2$ that cannot be rewritten as fractions, as well as transcendental numbers such as and );
[
]

$\backslash bold\; C$ or $\backslash mathbb\; C$, the set of all : , for example, .
[
]
Each of the above sets of numbers has an infinite number of elements. Each is a subset of the sets listed below it.
Sets of positive or negative numbers are sometimes denoted by superscript plus and minus signs, respectively. For example, $\backslash mathbf\{Q\}^+$ represents the set of positive rational numbers.
Functions
A function (or mapping) from a set to a set is a rule that assigns to each "input" element of an "output" that is an element of ; more formally, a function is a special kind of relation, one that relates each element of to exactly one element of . A function is called

injective (or onetoone) if it maps any two different elements of to different elements of ,

surjective (or onto) if for every element of , there is at least one element of that maps to it, and

bijective (or a onetoone correspondence) if the function is both injective and surjective — in this case, each element of is paired with a unique element of , and each element of is paired with a unique element of , so that there are no unpaired elements.
An injective function is called an injection, a surjective function is called a surjection, and a bijective function is called a bijection or onetoone correspondence.
Cardinality
The cardinality of a set , denoted , is the number of members of . For example, if , then . Repeated members in roster notation are not counted, so , too.
More formally, two sets share the same cardinality if there exists a onetoone correspondence between them.
The cardinality of the empty set is zero.
Infinite sets and infinite cardinality
The list of elements of some sets is endless, or
Infinite set. For example, the set
$\backslash N$ of
natural number is infinite.
[ In fact, all the special sets of numbers mentioned in the section above, are infinite. Infinite sets have infinite cardinality.
]
Some infinite cardinalities are greater than others. Arguably one of the most significant results from set theory is that the set of real number has greater cardinality than the set of natural numbers.
Sets with cardinality less than or equal to that of
$\backslash N$ are called
countable set; these are either finite sets or
countably infinite sets (sets of the same cardinality as
$\backslash N$); some authors use "countable" to mean "countably infinite". Sets with cardinality strictly greater than that of
$\backslash N$ are called
uncountable set.
However, it can be shown that the cardinality of a straight line (i.e., the number of points on a line) is the same as the cardinality of any line segment of that line, of the entire plane, and indeed of any finitedimensional Euclidean space.
The Continuum Hypothesis
The Continuum Hypothesis, formulated by Georg Cantor in 1878, is the statement that there is no set with cardinality strictly between the
alephnought and the cardinality of a straight line.
In 1963,
Paul Cohen proved that the Continuum Hypothesis is independent of the axiom system ZFC consisting of Zermelo–Fraenkel set theory with the axiom of choice.
[
]
(ZFC is the most widelystudied version of axiomatic set theory.)
Power sets
The power set of a set is the set of all subsets of .[ The empty set and itself are elements of the power set of , because these are both subsets of . For example, the power set of is . The power set of a set is commonly written as or .][
]
If has elements, then has elements. For example, has three elements, and its power set has elements, as shown above.
If is infinite (whether countable or uncountable), then is uncountable. Moreover, the power set is always strictly "bigger" than the original set, in the sense that any attempt to pair up the elements of with the elements of will leave some elements of unpaired. (There is never a bijection from onto .)
Partitions
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. That is, the subsets are pairwise disjoint (meaning any two sets of the partition contain no element in common), and the union of all the subsets of the partition is S.
Basic operations
There are several fundamental operations for constructing new sets from given sets.
Unions
Two sets can be joined: the union of and , denoted by , is the set of all things that are members of A or of B or of both.
Examples:
Some basic properties of unions:
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 is the set of all things that are members of both A and B. If then A and B are said to be disjoint.
Examples:
Some basic properties of intersections:
Complements
Two sets can also be "subtracted". The relative complement of B in A (also called the settheoretic difference of A and B), denoted by (or ), is the set of all elements that are members of A, but not members of B. It is valid to "subtract" members of a set that are not in the set, such as removing the element green from the set ; doing so will not affect the elements in the set.
In certain settings, all sets under discussion are considered to be subsets of a given universal set U. In such cases, is called the absolute complement or simply complement of A, and is denoted by A′ or A^{c}.
Examples:

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 include the following:
An extension of the complement is the symmetric difference, defined for sets A, B as
 $A\backslash ,\backslash Delta\backslash ,B\; =\; (A\; \backslash setminus\; B)\; \backslash cup\; (B\; \backslash setminus\; A).$
For example, the symmetric difference of and is the set . The power set of any set becomes a Boolean ring with symmetric difference as the addition of the ring (with the empty set as neutral element) and intersection as the multiplication of the ring.
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:
Some basic properties of Cartesian products:
Let A and B be finite sets; then the cardinality of the Cartesian product is the product of the cardinalities:

 A × B =  B × A =  A ×  B.
Applications
Sets are ubiquitous in modern mathematics. 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 in the construction of relations. A relation from a domain to a codomain is a subset of the Cartesian product . For example, considering the set of shapes in the game of the same name, the relation "beats" from to is the set ; thus beats in the game if the pair is a member of . Another example is the set of all pairs , where is real. This relation is a subset of , because the set of all squares is subset of the set of all real numbers. Since for every in , one and only one pair is found in , it is called a function. In functional notation, this relation can be written as .
Principle of inclusion and exclusion
The inclusion–exclusion principle is a counting technique that can be used to count the number of elements in a union of two sets—if the size of each set and the size of their intersection are known. It can be expressed symbolically as
 $A\; \backslash cup\; B\; =\; A\; +\; B\; \; A\; \backslash cap\; B.$
A more general form of the principle can be used to find the cardinality of any finite union of sets:
 $\backslash begin\{align\}$
\leftA_{1}\cup A_{2}\cup A_{3}\cup\ldots\cup A_{n}\right=& \left(\leftA_{1}\right+\leftA_{2}\right+\leftA_{3}\right+\ldots\leftA_{n}\right\right) \\
&{}  \left(\leftA_{1}\cap A_{2}\right+\leftA_{1}\cap A_{3}\right+\ldots\leftA_{n1}\cap A_{n}\right\right) \\
&{} + \ldots \\
&{} + \left(1\right)^{n1}\left(\leftA_{1}\cap A_{2}\cap A_{3}\cap\ldots\cap A_{n}\right\right).
\end{align}
De Morgan's laws
Augustus De Morgan stated two laws about sets.
If and are any two sets then,
The complement of union equals the complement of intersected with the complement of .
The complement of intersected with is equal to the complement of union to the complement of .
See also
Notes
External links