Set

:This article is about sets in mathematics. For other senses, see set (disambiguation).

Describing sets

Descriptions using words or lists

Not all sets have precise descriptions of any sort; they may simply be arbitrary collections, with no expressible "rule" saying what elements are in or out.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Some sets may be described in words, for example:

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:A is the set whose members are the first four positive whole numbers.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:B is the set whose members are the colors of the French flag.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

By convention, a set can also be defined by explicitly listing its elements between braces (sometimes called curly brackets or curly braces), for example:

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:C = {4, 2, 1, 3}

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:D = {red, white, blue}

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Notice that two different descriptions may define the same set. For example, for the sets defined above, A and C are identical, since they have precisely the same members. The shorthand A = C is used to express this equality. Similarly, for the sets defined above, B = D.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Set identity does not depend on the order in which the elements are listed, nor on whether there are repetitions in the list. For example, {6, 11} = {11, 6} = {11, 11, 6, 11}.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Descriptions using mathematical notation

For large sets (that is to say, sets in which there are many elements), it becomes highly impractical to explicitly write out the full list of contents. For example, E = {the first one thousand positive whole numbers} would, as a list, be as tedious to write as it would be to read. However, a mathematician would seldom describe E in words as above, preferring instead to use a symbolic shorthand:

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:E = {1, 2, 3, ..., 1000}

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

An abbreviated list can be used to describe a set such as E, where the elements can follow a pattern that is obvious to the reader. The full list is abbreviated using the ellipsis (...) symbol. When using this notation, care should also be taken to give enough elements to make the pattern clear. For example, the following set could, depending on the context, reasonably refer to either the first sixteen whole numbers or the first five powers of two:

Related Topics:
Ellipsis - Powers of two

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:X = {1, 2, ..., 16}

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

If, on the other hand, the characterizing property describes a less obvious pattern, then it is ill-advised to use an abbreviated list, which could serve to confuse the reader. For example, upon reading

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:F = {–4, –3, 0, ..., 357}

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

it is unclear that

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:F = {the first twenty numbers which are four less than a square number}.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

In such circumstances, mathematicians describe the characterizing property of the set using mathematical notation. For example:

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:F = {n^2 – 4 : n is a whole number and 0 leq n leq 19}

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

In this description, the colon (:) means such that, and the mathematician interprets this description as

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:F is the set of numbers of the form n^2 – 4, such that n is a whole number in the range from 0 to 19 inclusive. (Sometimes the pipe notation | is used instead of the colon.)

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

An explicit list of the contents of F can be found by evaluating the expression n^2 – 4 for each value of n from 0 to 19.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

For more information on describing sets see Set-builder notation.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Set membership

If something is or is not an element of a particular set then this is symbolised by in and otin respectively. So, for example:

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:*4 in A and 285 in F (since 285 = 17² − 4); but

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:*9 otin F and mathrm{green} otin B.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~