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   » Wiki: Set (mathematics)
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In , a set is a collection of distinct objects, considered as an object in its own right. For example, the numbers 2, 4, and 6 are distinct objects when considered separately, but when they are considered collectively they form a single set of size three, written {2,4,6}. The concept of a set is one of the most fundamental in mathematics. Developed at the end of the 19th century, is now a ubiquitous part of mathematics, and can be used as a foundation from which nearly all of mathematics can be derived. In mathematics education, elementary topics from set theory such as are taught at a young age, while more advanced concepts are taught as part of a university degree.

The German word Menge, rendered as "set" in English, was coined by in his work The Paradoxes of the Infinite.


Definition
A set is a well-defined collection of distinct objects. The objects that make up a set (also known as the set's elements or members) can be anything: numbers, people, letters of the alphabet, other sets, and so on. , 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:"Eine Menge, ist die Zusammenfassung bestimmter, wohlunterschiedener Objekte unserer Anschauung oder unseres Denkens – welche Elemente der Menge genannt werden – zu einem Ganzen."

Sets are conventionally denoted with . Sets A and B are equal if and only if they have precisely the same elements.

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 and the properties of sets are defined by a collection of . 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.


Describing sets
There are two ways of describing, or specifying the members of, a set. One way is by intensional definition, using a rule or semantic description:

A is the set whose members are the first four positive .
B is the set of colors of the French flag.

The second way is by extension – that is, listing each member of the set. An extensional definition is denoted by enclosing the list of members in :

C = {4, 2, 1, 3}
D = {blue, white, red}.

One often has the choice of specifying a set either intensionally or extensionally. In the examples above, for instance, A = C and B = D.

In an extensional definition, a set member can be listed two or more times, for example, {11, 6, 6}. However, per extensionality, two definitions of sets which differ only in that one of the definitions lists members multiple times define the same set. Hence, the set {11, 6, 6} is identical to the set {11, 6}. Moreover, the order in which the elements of a set are listed is irrelevant (unlike for a or ). We can illustrate these two important points with an example:

{6, 11} = {11, 6} = {11, 6, 6, 11} .

For sets with many elements, the enumeration of members can be abbreviated. For instance, the set of the first thousand positive integers may be specified extensionally as

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

where the ellipsis ("...") indicates that the list continues in the obvious way.

The notation with braces may also be used in an intensional specification of a set. In this usage, the braces have the meaning "the set of all ...". So, E = {playing card suits} is the set whose four members are A more general form of this is set-builder notation, through which, for instance, the set F of the twenty smallest integers that are four less than can be denoted

F = { n2 − 4 : n is an integer; and 0 ≤ n ≤ 19}.

In this notation, the colon (":") means "such that", and the description can be interpreted as " F is the set of all numbers of the form n2 − 4, such that n is a whole number in the range from 0 to 19 inclusive." Sometimes the ("|") is used instead of the colon.


Membership
If B is a set and x is one of the objects of B, this is denoted xB, and is read as "x belongs to B", or "x is an element of B". If y is not a member of B then this is written as yB, and is read as "y does not belong to B".

For example, with respect to the sets A = {1,2,3,4}, B = {blue, white, red}, and F = { n2 − 4 : n is an integer; and 0 ≤ n ≤ 19} defined above,

4 ∈ A and 12 ∈ F; but
9 ∉ F and green ∉ B.


Subsets
If every member of set A is also a member of set B, then A is said to be a subset of B, written AB (also pronounced A is contained in B). Equivalently, we can write BA, read as B is a superset of A, B includes A, or B contains A. The relationship between sets established by ⊆ is called inclusion or containment.

If A is a subset of, but not equal to, B, then A is called a proper subset of B, written AB ( A is a proper subset of B) or BA ( B is a proper superset of A).

The expressions AB and BA are used differently by different authors; some authors use them to mean the same as AB (respectively BA), whereas others use them to mean the same as AB (respectively BA).

Examples:

* The set of all men is a proper of the set of all people.
* {1, 3} ⊆ {1, 2, 3, 4}.
* {1, 2, 3, 4} ⊆ {1, 2, 3, 4}.

The is a subset of every set and every set is a subset of itself:

* ∅ ⊆ A.
* AA.

Every set is a subset of the :

* AU.

An obvious but useful identity, which can often be used to show that two seemingly different sets are equal:

* if and only if and .

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.


Power sets
The power set of a set S is the set of all subsets of S. The power set contains S itself and the empty set because these are both subsets of S. For example, the power set of the set {1, 2, 3} is

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