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'':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 '' well-defined'' collection of distinct elements, but problems arise from the vagueness of the term ''well-defined''.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 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.How sets are defined and set notation
Mathematical texts commonly denote sets by capital letters in italic, 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 curly brackets, 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 asInfinite 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 nonnegative integers is and the set of all integers isSemantic definition
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''.Set-builder notation
Set-builder 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: 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
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 ,The empty set
The ''empty set'' (or ''null set'') is the unique set that has no members. It is denoted or 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: * ∅ ⊆ ''A''. * ''A'' ⊆ ''A''.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. ) or blackboard bold (e.g. ) typeface. These include * or , the set of allFunctions
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 one-to-one) 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 one-to-one 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 ''one-to-one 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 one-to-one 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''. For example, the set of natural numbers 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 numbers has greater cardinality than the set of natural numbers. Sets with cardinality less than or equal to that of are called '' countable sets''; these are either finite sets or '' countably infinite sets'' (sets of the same cardinality as ); some authors use "countable" to mean "countably infinite". Sets with cardinality strictly greater than that of are called '' uncountable sets''. 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-dimensionalThe 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. In 1963, Paul Cohen proved that the continuum hypothesis isPower 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
Suppose that a universal set (a set containing all elements being discussed) has been fixed, and that is a subset of . * The complement of is the set of all elements (of ) that do ''not'' belong to . It may be denoted or . In set-builder notation, . The complement may also be called the ''absolute complement'' to distinguish it from the relative complement below. Example: If the universal set is taken to be the set of integers, then the complement of the set of even integers is the set of odd integers. Given any two sets and , * their union is the set of all things that are members of ''A'' or ''B'' or both. * their intersection is the set of all things that are members of both ''A'' and ''B''. If , then and are said to be ''disjoint''. * the set difference (also written ) is the set of all things that belong to but not . Especially when is a subset of , it is also called theApplications
Sets are ubiquitous in modern mathematics. For example, structures inPrinciple of inclusion and exclusion
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 also
* Algebra of sets * Alternative set theory * Category of sets * Class (set theory) * Dense set * Family of sets * Fuzzy set * Internal set * Mereology * Multiset * Principia Mathematica * Rough setNotes
References
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