Answer:

ANSWER AND SET

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Set

—A set in mathematics is a collection of well defined and distinct objects, considered as an object in its own right.

Sets are one of the most fundamental concepts in mathematics. Developed at the end of the 19th century, set theory 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 Venn diagrams are taught at a

young age, while more advanced concepts are taught as part of a university degree.

Content

1 Definition

2 Describing sets

3 Membership

3.1 Subsets

3.2 Power sets

4 Cardinality

5 Special sets

6 Basic operations

6.1 Unions

6.2 Intersections

6.3 Complements

6.4 Cartesian product

7 Applications

8 Axiomatic set theory

9 Principle of inclusion and exclusion

10 See also

11 Notes

12 References

13 External links

Definition

A set is a well defined collection of objects. Georg Cantor, the founder of set theory, gave the following definition of a set.

[1] A set is a gathering together into a whole of definite, distinct objects of our perception [Anschauung] and of our thought – which are called elements of the set.

The elements or members of a set can be anything: numbers, people, letters of the alphabet, other sets, and so on. Sets are conventionally denoted with capital letters.

—Sets A and B are equal if and only if they have precisely the same elements.

[2] As discussed below, the definition given above turned out to be inadequate for formal mathematics; instead, the notion of a "set" is taken as an undefined primitive in axiomatic set theory, and its properties are defined by the Zermelo–Fraenkel axioms.

The most basic properties are that a set

"has" elements, and that two sets are equal (one and the same) if and only if every element of one is an element of the other.

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 integers.

—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 curly brackets:

C = {4, 2, 1, 3}

D = {blue, white, red}.

Every element of a set must be unique;

no two members may be identical. (A multiset is a generalized concept of a set that relaxes this criterion.) All set operations preserve this property. The order in which the elements of a set or multiset are listed is irrelevant (unlike for a sequence or tuple).

Combining these two ideas into an example

{6, 11} = {11, 6} = {11, 6, 6, 11} because the extensional specification means merely that each of the elements listed is a member of the set.

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.