The intersection of two sets is made up of the objects contained in both sets, shown in a Venn diagram.

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Sets

In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. Although this appears to be a simple idea, sets are one of the most fundamental concepts in modern mathematics. The study of the structure of possible sets, set theory, is rich and ongoing. more...

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Having only been invented at the end of the 19th century, set theory is now a ubiquitous part of mathematics education, being introduced from primary school in many countries. Set theory can be viewed as the foundation upon which nearly all of mathematics can be derived.

Definition

At the beginning of his BeitrÃ¤ge zur BegrÃ¼ndung der transfiniten Mengenlehre, Georg Cantor, the principal creator of set theory, gave the following definition of a set:

The elements of a set, also called its members, can be anything: numbers, people, letters of the alphabet, other sets, and so on. Sets are conventionally denoted with capital letters. The statement that sets A and B are equal means that they have precisely the same members (i.e., every member of A is also a member of B and vice versa).

Unlike a multiset, every element of a set must be unique; no two members may be identical. All set operations preserve the property that each element of the set is unique. The order in which the elements of a set are listed is irrelevant, unlike a sequence or tuple.

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. See this example:

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 notated by enclosing the list of members in braces:

C = {4, 2, 1, 3}

D = {blue, white, red}

The order in which the elements of a set are listed in an extensional definition is irrelevant, as are any repetitions in the list. For example,

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

are equivalent, 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 whole numbers 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 even numbers can be written as {2, 4, 6, 8, ... }.