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]}

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]}

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]}

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 *BeitrÃ¤ge zur BegrÃ¼ndung der transfiniten Mengenlehre*:^{[13]}

A set is a gathering together into a whole of definite, distinct objects of our perception [Anschauung] or of our thoughtâ€”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 *BeitrÃ¤ge zur BegrÃ¼ndung der transfiniten Mengenlehre*:^{[13]}

A set is a gathering together into a whole of definite, distinct objects of our perception [Anschauung] or of our thoughtâ€”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 thoughtâ€”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]
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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]}

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: