Set (mathematics)

In mathematics, a set is a collection of distinct elements.[1][2][3] The elements that make up a set can be any kind of things: people, letters of the alphabet, numbers, points in space, lines, other geometrical shapes, variables, or even other sets.[4] Two sets are equal if and only if they have precisely the same elements.[5]

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.[4]

The concept of a set emerged in mathematics at the end of the 19th century.[6] The German word for set, Menge, was coined by Bernard Bolzano in his work Paradoxes of the Infinite.[7][8][9]

Georg Cantor, one of the founders of set theory, gave the following definition at the beginning of his Beiträge zur Begründung der transfiniten Mengenlehre:[10]

A set is a gathering together into a whole of definite, distinct objects of our perception or our thought—which are called elements of the set.

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 a member of B, and every element of B is an element of A; this property is called the extensionality of sets.[11]

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:

Naïve set theory defines a set as any well-defined collection of distinct elements, but problems arise from the vagueness of the term well-defined.

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.[12] 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 first-order logic. According to Gödel's incompleteness theorems however, it is not possible to use first-order logic to prove any such particular axiomatic set theory is free from paradox.[citation needed]

Mathematical texts commonly denote sets by capital letters[13][4][14] in italic, such as A, B, C.[14][15] A set may also be called a collection or family, especially when its elements are themselves sets.

One way to define a set is to use a rule to determine what the elements are:

Roster or enumeration notation defines a set by listing its elements between curly brackets, separated by commas:[18][19][20][21]

For sets with many elements, especially those following an implicit pattern, the list of members can be abbreviated using an ellipsis "...".[24][25] For instance, the set of the first thousand positive integers may be specified in roster notation as

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 nonnegative integers is

Set-builder notation specifies a set as a selection from a larger set, determined by a condition on the elements.[17][26][27] For example, a set F can be defined as follows:

In this notation, the vertical bar "|" means "such that", and the description can be interpreted as "F is the set of all numbers n such that n is an integer in the range from 0 to 19 inclusive". Some authors use a colon ":" instead of the vertical bar.[28]

If B is a set and x is an element of B, this is written in shorthand as xB, which can also be read as "x belongs to B", or "x is in B".[11] The statement "y is not an element of B" is written as yB, which can also be read as or "y is not in B".[29][14][30]

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 AB,[34] or BA.[35][14] 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: AB and BA is equivalent to A = B.[26]

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 AB. Likewise, BA 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 AB and BA to mean A is any subset of B (and not necessarily a proper subset),[36][29] while others reserve AB and BA for cases where A is a proper subset of B.[34]

The empty set is a subset of every set,[31] and every set is a subset of itself:[36]

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 A is a subset of B, then the region representing A is completely inside the region representing B. If two sets have no elements in common, the regions do not overlap.

A Venn diagram, in contrast, is a graphical representation of n sets in which the n loops divide the plane into 2n zones such that for each way of selecting some of the n 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 A, B, and C, there should be a zone for the elements that are inside A and C and outside B (even if such elements do not exist).

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.

Each of the above sets of numbers has an infinite number of elements. Each is a subset of the sets listed below it.

A function (or mapping) from a set A to a set B is a rule that assigns to each "input" element of A an "output" that is an element of B; more formally, a function is a special kind of relation, one that relates each element of A to exactly one element of B. A function is called

An injective function is called an injection, a surjective function is called a surjection, and a bijective function is called a bijection or one-to-one correspondence.

More formally, two sets share the same cardinality if there exists a one-to-one correspondence between them.

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 segment of that line, of the entire plane, and indeed of any finite-dimensional Euclidean space.[43]

The Continuum Hypothesis, formulated by Georg Cantor in 1878, is the statement that there is no set with cardinality strictly between the cardinality of the natural numbers and the cardinality of a straight line.[44] 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.[45] (ZFC is the most widely-studied version of axiomatic set theory.)

The power set of an infinite (either countable or uncountable) set is always uncountable. Moreover, within the most widely-used frameworks of set theory, the power set of a set is always strictly "bigger" than the original set, in the sense that there is no way to pair every element of S with exactly one element of P(S). (There is never an onto map or surjection from S onto P(S).)[48]

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.[49][50]

There are several fundamental operations for constructing new sets from given sets.

Two sets can be "added" together. The union of A and B, denoted by A ∪ B,[14] is the set of all things that are members of either A or B.

A new set can also be constructed by determining which members two sets have "in common". The intersection of A and B, denoted by AB,[14] is the set of all things that are members of both A and B. If AB = ∅, then A and B are said to be disjoint.

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′ or Ac.[14]

An extension of the complement is the symmetric difference, defined for sets A, B as

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,[14] is the set of all ordered pairs (a, b) such that a is a member of A and b is a member of B.

Let A and B be finite sets; then the cardinality of the Cartesian product is the product of the cardinalities:

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.

The inclusion-exclusion principle is used to calculate the size of the union of sets: the size of the union is the size of the two sets, minus the size of their intersection.

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 more general form of the principle can be used to find the cardinality of any finite union of sets:

The complement of A union B equals the complement of A intersected with the complement of B.

The complement of A intersected with B is equal to the complement of A union to the complement of B.