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   » Wiki: Set (mathematics)
Tag Wiki 'Set (mathematics)'.

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 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.

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 set members multiple times, define, in fact, the same set. Hence, the set {11, 6, 6} is exactly 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. Ellipses may also be used where sets have infinitely many members. Thus the set of positive can be written as

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.

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.

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).


* 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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