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In mathematics, a set is a well-defined collection of distinct objects, considered as an object in its own right.[1][2] The arrangement of the objects in the set does not matter. A set may be denoted by placing its objects between a pair of curly braces. For example, the numbers 2, 4, and 6 are distinct objects when considered separately; when considered collectively, they form a single set of size three, written as {2, 4, 6}, which could also be written as {2, 6, 4}, {4, 2, 6}, {4, 6, 2}, {6, 2, 4} or {6, 4, 2}.[3] Sets can also be denoted using capital roman letters in italic such as , , .[4][5]

The concept of a set is one of the most fundamental in mathematics.[6] Developed at the end of the 19th century,[7] the theory of sets is now a ubiquitous part of mathematics, and can be used as a foundation from which nearly all of mathematics can be derived.[6]

Etymology

The German word Menge, rendered as "set" in English, was coined by Bernard Bolzano in his work The Paradoxes of the Infinite.[8][9][10]

Definition

Passage with a translation of the original set definition of Georg Cantor. The German word Menge for set is translated with aggregate here.

A set is a well-defined collection of distinct objects.[1][2] The objects that make up a set (also known as the set's elements or members)[6] Developed at the end of the 19th century,[7] the theory of sets is now a ubiquitous part of mathematics, and can be used as a foundation from which nearly all of mathematics can be derived.[6]

The German word Menge, rendered as "set" in English, was coined by Bernard Bolzano in his work The Paradoxes of the Infinite.[8][9][10]

Definition

Passage with a translation of the original set definition of Georg Cantor. The German word Menge for set is translated with aggregate here.

A set is a well-defined collection of distinct objects.[1][2] The objects that make up a set (also known as the set's elements or members)[11] can be anything: numbers, people, letters of the alphabet, other sets, and so on.[12] Georg Cantor, one of the founders of set theory, gave the following definition of a set at the beginning of his Beiträge zur Begründung der transfiniten Mengenlehre:[13]

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.[14][15][4] Sets A and B are equal if and only if they have precisely the same elements.[16]

For technical reasons, Cantor's definition turned out to be inadequate; today, in contexts where more rigor is required, one can use axiomatic set theory, in which the notion of a "set" is taken as a primitive notion, and the properties of sets are defined by a collection of axioms.[17] The most basic properties are that a set can have elements, and that two sets are equal (one and the same) if and only if every element of each set is an element of the other; this prop

A set is a well-defined collection of distinct objects.[1][2] The objects that make up a set (also known as the set's elements or members)[11] can be anything: numbers, people, letters of the alphabet, other sets, and so on.[12] Georg Cantor, one of the founders of set theory, gave the following definition of a set at the beginning of his Beiträge zur Begründung der transfiniten Mengenlehre:[13]

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.[14][15][4] Sets A and B are equal if and only if they have precisely the same elements.[16]

For technical reasons, Cantor's definition turned out to be inadequate; today, in contexts where more rigor is required, one can use axiomatic set theory, in which the notion of a "set" i

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.[14]capital letters.[14][15][4] Sets A and B are equal if and only if they have precisely the same elements.[16]

For technic

For technical reasons, Cantor's definition turned out to be inadequate; today, in contexts where more rigor is required, one can use axiomatic set theory, in which the notion of a "set" is taken as a primitive notion, and the properties of sets are defined by a collection of axioms.[17] The most basic properties are that a set can have elements, and that two sets are equal (one and the same) if and only if every element of each set is an element of the other; this property is called the extensionality of sets.[18]

There are two common ways of describing or specifying the members of a set: roster notation and set builder notation.[19][20] These are examples of extensional and intensional definitions of sets, respectively.[21]

Roster notation

The Roster notation (or enumeration notation) method of defining a set consist of listing each member of the set.[19][22][23] More specifically, in roster notation (an example of extensional definition),[21] the set is denoted by enclosing the list of members in curly brackets:

A = {4, 2, 1, 3}
B = {blue, white, red}.

For sets with many elements, the enumeration of members can be abbreviated.[24]For sets with many elements, the enumeration of members can be abbreviated.[24][25] For instance, the set of the first thousand positive integers may be specified in roster notation as

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

where the ellipsis ("...") indicates that the list continues in according to the demonstrated pattern.[24]

In roster notation, listing a member r

In roster notation, listing a member repeatedly does not change the set, for example, the set {11, 6, 6} is identical to the set {11, 6}.[26][failed verification] Moreover, the order in which the elements of a set are listed is irrelevant (unlike for a sequence or tuple), so {6, 11} is yet again the same set.[26][5]

In set-builder notation, the set is specified as a selection from a larger set, determined by a condition involving the elements.[27][28] For example, a set F can be specified as follows: