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Set
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The Set domain allows one to represent explicit finite sets of values.
These are similar to lists, but duplicate elements are not allowed.

Sets can be created by giving a fixed set of values ::

  s := set [x**2-1, y**2-1, z**2-1]
      2      2      2
    {x  - 1,y  - 1,z  - 1}
                           Type: Set Polynomial Integer

or by using a collect form, just as for lists.  In either case, the
set is formed from a finite collection of values::

  t := set [x**i - i+1 for i in 2..10 | prime? i]
      2      3      5      7
    {x  - 1,x  - 2,x  - 4,x  - 6}
                           Type: Set Polynomial Integer

The basic operations on sets are intersect, union, difference, and
symmetricDifference. ::

  i := intersect(s,t)
      2
    {x  - 1}
                           Type: Set Polynomial Integer

  u := union(s,t)
      2      3      5      7      2      2
    {x  - 1,x  - 2,x  - 4,x  - 6,y  - 1,z  - 1}
                           Type: Set Polynomial Integer

The set difference(s,t) contains those members of s which are not in t. ::

  difference(s,t)
      2      2
    {y  - 1,z  - 1}
                           Type: Set Polynomial Integer

The set symmetricDifference(s,t) contains those elements which are
in s or t but not in both. ::

  symmetricDifference(s,t)
      3      5      7      2      2
    {x  - 2,x  - 4,x  - 6,y  - 1,z  - 1}
                           Type: Set Polynomial Integer

Set membership is tested using the member? operation. ::

  member?(y, s)
    false
                           Type: Boolean

  member?((y+1)*(y-1), s)
    true
                           Type: Boolean

The subset? function determines whether one set is a subset of another. ::

  subset?(i, s)
    true
                           Type: Boolean

  subset?(u, s)
    false
                           Type: Boolean

When the base type is finite, the absolute complement of a set is
defined.  This finds the set of all multiplicative generators of 
PrimeField 11---the integers mod 11. ::

  gs := set [g for i in 1..11 | primitive?(g := i::PF 11)] 
    {2,6,7,8}
                           Type: Set PrimeField 11

The following values are not generators. ::

  complement gs 
    {1,3,4,5,9,10,0}
                           Type: Set PrimeField 11

Often the members of a set are computed individually; in addition,
values can be inserted or removed from a set over the course of a
computation.

There are two ways to do this: ::

  a := set [i**2 for i in 1..5]
    {1,4,9,16,25}
                           Type: Set PositiveInteger

One is to view a set as a data structure and to apply updating operations. ::

  insert!(32, a)
    {1,4,9,16,25,32}
                           Type: Set PositiveInteger

  remove!(25, a)
    {1,4,9,16,32}
                           Type: Set PositiveInteger

  a
    {1,4,9,16,32}
                           Type: Set PositiveInteger

The other way is to view a set as a mathematical entity and to
create new sets from old. ::

  b := b0 := set [i**2 for i in 1..5]
    {1,4,9,16,25}
                           Type: Set PositiveInteger

  b := union(b, {32})
    {1,4,9,16,25,32}
                           Type: Set PositiveInteger

  b := difference(b, {25})
    {1,4,9,16,32}
                           Type: Set PositiveInteger

  b0
    {1,4,9,16,25}
                           Type: Set PositiveInteger

See Also:

* )help List
* )show Set