Greg
Ordering provides uniqueness. The names
set 1;0;2 and set 1;1;2;0 etc etc all refer to the same j object
< 0;1;2
This parallels the fact that in math the sets { 1, 0, 2 } and ( 1, 1, 2, 0 }
ARE THE SAME SET.
Uniqueness permits us to use to use Match -: to test whether two possibly
different sets are the same set, for example
(A un1 B) -: (A un2 B)
to test whether two definitions of union are equivalent.
Ordering also makes possible my very short definition of verb isset which tests
whether a J array is a set.
My objective is to have a treatment of sets that parallels mathematical sets.
For some applications that may not be the best plan, but I know the way
mathematical sets work, and as Darrell Royal of University of Texas football
fame used to say, "I dance with who brung me."
--Kip
Sent from my iPad
> On Nov 16, 2013, at 2:47 PM, greg heil <[email protected]> wrote:
>
> Kip
>
>> i do want to apologize for a duplicate question, i saw too late it had
>> already been asked. Thanks for the elaboration. Still i do not see in your
>> elaboration, which seems primarily about enclosing, a justification for the
>> ordering of sets, while, eg, nub is not? Sorting in sets seems primarily an
>> algorithmic not mathematical issue.
>
> greg
> ~krsnadas.org
>
> --
>
> from: km <[email protected]> via forums.jsoftware.com
> to: "[email protected]" <[email protected]>
> date: 16 November 2013 12:00
> subject: Re: [Jprogramming] Powersets (was RE: Sets)
>
>> OK, now: but the discussion is mathematical and philosophical and maybe
>> belongs in Chat.
>
>> In math the notations { 0, 1, 2 } , { 1, 2, 0 } , and { 1, 0, 2, 1 } all
>> refer to THE set whose elements are 0 and 1 and 2 . There is no order in
>> this set, although there is order in notations naming the set. Also, an
>> element cannot belong to a set more than once, although the same element may
>> be named more than once in the notation. Once you know the elements are the
>> numbers 0 and 2 and 1 you know what set you are talking about. There are
>> things about this set we do not care about. Is it blue? Is it left-handed?
>> We don't care. The set is a unique mathematical object, but as is common in
>> math, we don't care even what the object IS , we only care that it is a set,
>> whatever that is, and that its elements are the three numbers 2 and 0 and 1.
>
>> In a J implementation we DO care what J object is representing a set, and we
>> would like the J representation to have properties matching the mathematical
>> properties. So given my definition
>
> set =: [: < [: /:~ ~.
>
>> and the understanding it is to be applied to a list of boxes, we find that
>> the notations set 0;1;2 and set 1;2;0 and set 1;0;2;1 all represent the same
>> J object, namely < 0;1;2 . With my definition < 0;1;2 is THE J object
>> representing THE mathematical set whose elements are 1 and 2 and 0 .
>
>> Why do I have an enclosing box for the set and for each element in the
>> definitions "a set is a box enclosing a sorted list of boxes", and "an
>> element is the contents of a box in the sorted list" ?
>
>> A fair reason for the outer enclosing box is that in math a list and a set
>> are different. A mathematical list can have duplicates, a mathematical set
>> cannot. An order is a required part of a mathematical list, but not of a
>> mathematical set. Because of the outer enclosing box, my J sets are atoms,
>> not lists.
>
>> A good reason (in my opinion) for the outer enclosing box is that with my J
>> definition the empty set is a normal set, namely set '' , a box enclosing an
>> empty list. The empty list has no boxes and hence the empty set has no
>> elements.
>
>> A psychological reason is that the outer enclosing box comfortingly
>> resembles the enclosing braces in the mathematical notation.
>
>> Why are elements boxed? That permits any J array to be an element. The
>> elements of set 4 ; i. 2 2 are the number 4 and the 2 by 2 matrix i. 2 2 .
>
> --Kip Murray
>
> --
>
> from: greg heil <[email protected]>
> to: Programming forum <[email protected]>
> date: 16 November 2013 09:14
> subject: Re: [Jprogramming] Powersets (was RE: Sets)
>
> Perhaps now?
> Several times you have asked that revisions including sort be made ...
> yet my understanding of the classical definition of a set is
> _unordered_...?
>
> greg
> ~krsnadas.org
>
> --
>
> from: km <[email protected]>
> to: "[email protected]" <[email protected]>
> date: 16 November 2013 08:57
> subject: Re: [Jprogramming] Powersets (was RE: Sets)
>
>> ...Why do I include Sort /:~ in my proposal? Let's discuss that another
>> time...
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