I have revised my model of set theory so that every array, including i.0 0, can
be an element, and have adapted Raul's verb powSet to this model.
In this model 2 is not a set, but you can define a set corresponding to the
natural number 2 of mathematical set theory.
If I can learn to adapt Fraser's verb sortSet to this model, I may be able to
remove its current restriction that the curtail of a "set" must be sorted and
have no duplicates. I am returning to this model because I want a unique empty
set, and I do not want any sets which are elements of themselves.
]E =: , < i.0 0 NB. empty set
++
++
]M =: 'a';'b';E NB. simple sets can be defined this way.
+-+-++
|a|b||
+-+-++
NB. above, note "set-marker" tail
NB. following verb pwrSet is adapted from Raul's powSet, see Details below:
pwrSet M
+--+----+------+----++
|++|+-++|+-+-++|+-++||
|++||a||||a|b||||b||||
| |+-++|+-+-++|+-++||
+--+----+------+----++
NB. Imitating mathematical set theory
]zero =: E NB. empty set
++
++
]one =: zero;E NB. set whose only element is zero
+--++
|++||
|++||
+--++
]two =: zero;one;E NB. set whose only elements are zero and one
+--+-----++
|++|+--++||
|++||++||||
| ||++||||
| |+--++||
+--+-----++
]singletontwo =: two;E NB. set whose only element is two
+-----------++
|+--+-----++||
||++|+--++||||
||++||++||||||
|| ||++||||||
|| |+--++||||
|+--+-----++||
+-----------++
countset two
2
countset singletontwo
1
two issameset singletontwo
0
isset two
1
isset 2
0
Details
NB. A "set" is a list of boxes with tail <i.0 0 and having a sorted curtail
NB. with no duplicates. An "element" is the open of a box in the curtail.
NB. The tail <i.0 0 is a "set-marker" which permits the list ,<i.0 0
NB. to be the _unique_ empty set.
issequence =: (1 = #...@$) *. (0 < L.)
isset =: issequence *. ( (< i.0 0) -: {: ) *. ( (-: /:~)*.(-: ~.) )@}:
Empty =: , < i.0 0 NB. empty set
E =: Empty
set =: ( /:~...@~.@}: , {: ) : [:
NB. usage set list-of-boxes-ending-with-Empty creates a set;
NB. elements are opens of boxes in the curtail of the argument
issameset =: -:
is =: issameset
iselementof =: <@[ e. }:@]
in =: iselementof
countset =: #...@}:
index =: }:@[ i. <@] NB. find index of element
indexof =: index
get =: [:`({:: }:)@.(1 > l...@[) NB. retrieve element
u =: [: set -. , ] NB. union
less =: Empty ,~ -. NB. difference
n =: [ less less NB. intersection
issubsetof =: [ is n
part =: issubsetof
isemptyset =: -: Empty"_
pwrset =: 3 : 0 NB. adapted from Fraser's explicit version
if.
isemptyset y
do.
Empty;Empty NB. set whose only element is Empty
else.
(Empty ,~ /:~) (#&y) &.> ( <"1 ] 1 ,~"1 (#: i.2^<:#y) )
end.
)
NB. pwrSet below is adapted from Raul's powSet
pwrSet =: (<i.0 0) ,~ ] /:~@:(<@#~) 1 (,~"1) 2 (#"1~ {:)@#:@i...@^ #...@}:
Kip Murray wrote:
> This has to do with Raul's question,
>
> Raul Miller wrote:
> > . . . how do you distinguish between an element of a
> > set and a set containing only that element?
>
>
> I will discuss three version of sets. Bear with me, we will get to "2 is the
> set whose only element is 2" in the second version.
>
>
> In mathematical set theory,
>
> 2 looks different from {2} -- but how do you know 2 isn't {2}? The
> question
> isn't as fanciful as it looks, for it is common in set theory to say
>
> 0 is defined to be {}, the empty set
> 1 is defined to be {0}, the set whose only element is 0
> 2 is defined to be {0,1}, the set whose only elements are 0 and 1
> ... and so on: the natural numbers are sets, and how do we know set 2 isn't
> the
> set {2}?
>
> W see 0 is {}, 1 is { {} }, and
>
> 2 is { {} , { {} } }, whereas
> {2} is { { {} , { {} } } }, and we are sure 2 is not {2}. (For one thing,
> {2}
> has only one element, and 2 has two elements!)
>
>
> But in the current J model of sets we have
>
> <"_1 ] 2 NB. <"_1 boxes items
> +-+
> |2|
> +-+
>
> which means 2 is the only element of set 2 -- we have defined sets to be open
> or
> boxed arrays and the elements of open arrays to be their items.
>
>
> In my original J model of sets,
>
> 2 NB. looks different from (set ,<2) the set whose only element
> is 2
> 2
> set ,<2 NB. argument is required to be a list of boxes containing
> elements
> +-++
> |2||
> +-++
>
> count set <2 NB. number of elements
> 1
> 2 iselement set <2
> 1
> 2 -: set <2 NB. not only looks different, is different
> 0
>
>
> Reference material:
>
>
> Current implementor catechism
>
> Question 1: What is a set?
> Answer: A set is an array.
>
> Question 2: What is an element of a set?
> Answer: If the set is open, an element is an item of the set. If the set
> is
> boxed, an element is the open of an atom of the set.
>
>
> Earlier implementor catechism (revised)
>
> Question 0: What is a sequence?
> Answer: A "sequence" is a list of boxes. A "term" of a sequence is the
> open
> of one of its boxes.
>
> Question 1: What is a set?
> Answer: A "set" is a sequence whose last term is i.0 0 .
>
> Question 2: What is an element of a set?
> Answer: An "element" of a set is a term of its curtail.
>
> ----------------------------------------------------------------------
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