In mathematics, a set is a collection of different things;[1] [2] [3] these things are called elements or members of the set and are typically mathematical objects of any kind: numbers, symbols, points in space, lines, other geometrical shapes, variables, or even other sets. A set may have a finite number of elements or be an infinite set. There is a unique set with no elements, called the empty set; a set with a single element is a singleton.
Sets are uniquely characterized by their elements; this means that two sets that have precisely the same elements are equal (they are the same set).[4] This property is called extensionality. In particular, this implies that there is only one empty set.
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.
Mathematical texts commonly denote sets by capital letters[5] in italic, such as,, .[6] A set may also be called a collection or family, especially when its elements are themselves sets.
Roster or enumeration notation defines a set by listing its elements between curly brackets, separated by commas:[7] [8] [9] [10]
This notation was introduced by Ernst Zermelo in 1908.[11] 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.[12] [13]
For sets with many elements, especially those following an implicit pattern, the list of members can be abbreviated using an ellipsis ''.[14] [15] 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 isand the set of all integers is
Another way to define a set is to use a rule to determine what the elements are:
Such a definition is called a semantic description.[16]
See main article: Set-builder notation.
Set-builder notation specifies a set as a selection from a larger set, determined by a condition on the elements.[16] [17] [18] For example, a set can be defined as follows:
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.[19]
Philosophy uses specific terms to classify types of definitions:
See main article: Element (mathematics). 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".[20] [21]
For example, with respect to the sets,, and,
See main article: Empty set. The empty set (or null set) is the unique set that has no members. It is denoted,
\emptyset
See main article: Singleton (mathematics). A singleton set is a set with exactly one element; such a set may also be called a unit set.[4] 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.
See main article: Subset. 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,[25] or .[26] 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: and is equivalent to A = B.[17]
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 . Likewise, 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 and to mean A is any subset of B (and not necessarily a proper subset),[20] while others reserve and for cases where A is a proper subset of B.[25]
Examples:
The empty set is a subset of every set, and every set is a subset of itself:
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).
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.
Z
Z
N
N
N=\{0,1,2,3,...\}
Z
Z
Z=\{...,-2,-1,0,1,2,3,...\}
Q
Q
|
a,b\inZ,b\ne0\right\}
R
R
\sqrt2
C
C
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,
Q+
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
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.
See main article: Cardinality.
The cardinality of a set, denoted, is the number of members of .[28] For example, if, then . Repeated members in roster notation are not counted,[29] [30] so, too.
More formally, two sets share the same cardinality if there exists a bijection between them.
The cardinality of the empty set is zero.[31]
The list of elements of some sets is endless, or infinite. For example, the set
\N
Some infinite cardinalities are greater than others. Arguably one of the most significant results from set theory is that the set of real numbers has greater cardinality than the set of natural numbers.[32] Sets with cardinality less than or equal to that of
\N
\N
\N
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.[33]
See main article: Continuum hypothesis. 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.[34] 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.[35] (ZFC is the most widely-studied version of axiomatic set theory.)
See main article: Power set. 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 .)[36]
See main article: Partition of a set.
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.[37]
See main article: Algebra of sets.
Suppose that a universal set (a set containing all elements being discussed) has been fixed, and that is a subset of .
Ac=\{a\inU:a\notinA\}
Given any two sets and,
A\DeltaB=(A\setminusB)\cup(B\setminusA)
Examples:
The operations above satisfy many identities. For example, one of De Morgan's laws states that (that is, the elements outside the union of and are the elements that are outside and outside).
The cardinality of is the product of the cardinalities of and .(This is an elementary fact when and are finite. When one or both are infinite, multiplication of cardinal numbers is defined to make this true.)
The power set of any set becomes a Boolean ring with symmetric difference as the addition of the ring and intersection as the multiplication of the ring.
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 .
See main article: Inclusion–exclusion principle.
The inclusion–exclusion principle is a technique for counting the elements in a union of two finite sets in terms of the sizes of the two sets and their intersection. It can be expressed symbolically as
A more general form of the principle gives the cardinality of any finite union of finite sets:
See main article: Set theory. The concept of a set emerged in mathematics at the end of the 19th century.[38] The German word for set, Menge, was coined by Bernard Bolzano in his work Paradoxes of the Infinite.[39] [40] [41] Georg Cantor, one of the founders of set theory, gave the following definition at the beginning of his Beiträge zur Begründung der transfiniten Mengenlehre:[42] [43]
Bertrand Russell introduced the distinction between a set and a class (a set is a class, but some classes, such as the class of all sets, are not sets; see Russell's paradox):[44]
See main article: 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. As a consequence, e.g. and represent the same set. Unlike sets, multisets can be distinguished by the number of occurrences of an element; e.g. and represent different multisets, while and are equal. Tuples can even be distinguished by element order; e.g. and represent different tuples.
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.[45] 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.[46]