First, we become familiar with nouns.

Definition: A list, with individual entries separated by commas, is called a SET when it is enclosed in brackets { }.

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• (orderings and repetitions are considered irrelevant)
• (The brackets turn an ordinary list of things into a mathematical entity!)

Example:

• "king, silver, pink, red" is a list of salmon types
• {king, silver, pink, red} is a SET which contains the names of some salmon types.
• Alternative / equivalent ways to write this set include:

  {king, silver, red, pink}

  {king, pink, silver, red}

  {king, pink, red, silver}

  {king, red, silver, pink}

  {king, red, pink, silver}

• Can you find another 18 equivalent (but "different") ways to write this?

Second, we find verbs that work with the nouns we know.

• Operations: union, intersection, complement
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  • These give us something to do with sets, a legitimate algebraic game without needing any numbers.

   

• Analyses:
• names for sets

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  • When there are a bunch around we give them names; this common human solution to avoid confusion is also a very, very popular mathematical method.

• numeric measure / description of a sets

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  • The number of elements in a set is the most natural measure though we are often concerned with a percentage of the whole.

• pieces and parts of sets (called SUBSETS)

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  • Often we are concerned only with selected individuals from a set.

• order the elements of sets

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  • This is popular among computers and word processors, which allows us to add "concatenate" as an operation.

Example:

Let {king, silver, pink, red} be called the set Salmon.

• Use S as a nickname for the set called Salmon.
  Then we write S = {king, silver, pink, red}
• The number of elements in S is easily counted to be 4, and is written mathematically as n(S) = 4.
• Let {pink, silver, red} = C. Then C is a subset of S, n(C) = 3, and C is three fourths the size of S

A phone survey of Ketchikan draws from the set of all names in the Ketchikan phone book.

• This set is then called the DOMAIN or REALM OF ACTION or UNIVERSAL SET for the operation called a survey.
• (This domain is often ordered by an alphabet.)

A function being graphed on the coordinate plane draws from the set of all real numbers.

• This set, commonly known as R , is called the DOMAIN or REALM OF ACTION or UNIVERSAL SET for the operation called a function.
• (This domain is often ordered by the notion >.)

A computer reading a HD floppy disc draws from the set of all information bytes stored on the disc.

• This set is then called the DOMAIN or REALM OF ACTION or UNIVERSAL SET for the operation called read the disc.
• (This domain is ordered by a matrix assignment.)

These three (important mathematical applications of) sets are very large and unwieldy for novices.

They are also encumbered by the "natural" orderings commonly associated with them.

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• For introductory classroom purposes of hands-on pattern-development and demonstration,
• to construct the properties and concepts of set theory, which will enable future work with large, complex sets
• there is a common, mathematically and historically rich, manipulative available.