It's a valid question. There were technical reasons for putting a: at the end
of every set. One was, I wanted every set to satisfy
isboxed =: 0 < L.
Uniformity keeps the theory simple, and for me this is a theory exercise.
Let's say
]M =: 1;2
┌─┬─┐
│1│2│
└─┴─┘
then
(0$a:) -: M -. M
1
that is, M -. M appropriately matches your proposed empty set. But
L. M -. M
0
so the operation -. has led to a not boxed result, and of course your proposed
empty set also is not boxed. My use of a: provides the uniformity I wanted.
As a practical matter, once the few fundamental definitions are written, we can
use them without thinking about a: or the fact E =: ,a: -- and I think the
display
E
┌┐
││
└┘
is a bonus. The joke about the Chesire-Cat helps with the reading. Carroll
was
a mathematician, and I wonder whether he was thinking about the empty set!
Kip
On 10/10/2010 4:57 PM, Marshall Lochbaum wrote:
> But does it serve a purpose? When every single verb you have has to be
> adjusted for the end of the set not being an element, it doesn't really make
> much sense to keep a: .
>
> Marshall
>
> -----Original Message-----
> From: [email protected]
> [mailto:[email protected]] On Behalf Of Kip Murray
> Sent: Sunday, October 10, 2010 5:15 PM
> To: Programming forum
> Subject: Re: [Jprogramming] Sets
>
> The Ace a: is like the grin on Lewis Carroll's Chesire-Cat. When everything
> disappears except the grin, you are looking at the empty set.
>
>
> On 10/10/2010 11:51 AM, Marshall Lochbaum wrote:
>> I don't see the usefulness of the a: at the end of the sets. An empty set
>> should just be represented by 0$a:, and is clearly identifiable because it
>> gives no output.
>>
>> Marshall
>>
>> -----Original Message-----
>> From: [email protected]
>> [mailto:[email protected]] On Behalf Of Kip Murray
>> Sent: Sunday, October 10, 2010 12:02 AM
>> To: Programming forum
>> Subject: [Jprogramming] Sets
>>
>> Here is my latest attempt to model set theory in J. All sets have distinct
>> elements and are ordered by /:~ so that match -: determines whether two sets
>> are the same. Sets must be created by the verb set or by provided
>> operations. The intention is theoretical not practical! --Kip Murray
>>
>> ]A =: set 0;'b';2 NB. elements 0 'b' 2 are put in boxes preceding the
>> last ┌─┬─┬─┬┐ │0│2│b││ └─┴─┴─┴┘
>> ]B =: set 2;'b';'b';0 NB. same elements so same set ┌─┬─┬─┬┐ │0│2│b││
>> └─┴─┴─┴┘
>>
>> A -: B
>> 1
>>
>> ]C =: set 2;'b';'c';'d'
>> ┌─┬─┬─┬─┬┐
>> │2│b│c│d││
>> └─┴─┴─┴─┴┘
>>
>> B sand C NB. intersection, "set and"
>> ┌─┬─┬┐
>> │2│b││
>> └─┴─┴┘
>> B sor C NB. union
>> ┌─┬─┬─┬─┬─┬┐
>> │0│2│b│c│d││
>> └─┴─┴─┴─┴─┴┘
>>
>> (A sand B sor C) -: (A sand B) sor (A sand C) NB. distributive law
>> 1
>>
>> pwrset A NB. A has 3 elements, power set has 2^3, including the empty
>> set ┌──────┬────────┬────┬──────┬────┬──────┬──┬────┬┐
>> │┌─┬─┬┐│┌─┬─┬─┬┐│┌─┬┐│┌─┬─┬┐│┌─┬┐│┌─┬─┬┐│┌┐│┌─┬┐││
>> ││0│2││││0│2│b││││0││││0│b││││2││││2│b│││││││b││││
>> │└─┴─┴┘│└─┴─┴─┴┘│└─┴┘│└─┴─┴┘│└─┴┘│└─┴─┴┘│└┘│└─┴┘││
>> └──────┴────────┴────┴──────┴────┴──────┴──┴────┴┘
>> NB. Elements are contained in boxes preceding the last which is always
>> NB. the Boxed Empty a: (Ace). The use of a: permits a unique and
>> visible
>> NB. empty set, viz
>>
>> (,a:) -: E =: A less A NB. see verb less below
>> 1
>> E
>> ┌┐
>> ││
>> └┘
>> a:
>> ┌┐
>> ││
>> └┘
>> E -: a:
>> 0
>>
>> NB. Definitions
>>
>> E =: ,a: NB. empty set
>> set =: a: ,~ [: /:~ ~. NB. create set from boxed list y
>> NB. each box of y encloses an element
>> get =: { }: NB. get boxed elements (from curtail because
>> NB. elements are inside boxes of curtail)
>> isin =: e. }: NB. Do boxes in list x contain elements of y?
>> less =: a: ,~ -.&}: NB. remove elements of y from x
>> sand =: [ less less NB. intersection, "set and"
>> sor =: a: ,~ [: /:~ [: ~. ,&}: NB. union, "set or"
>> diff =: less sor less~ NB. symmetric difference
>> card =: [: # }: NB. count elements: cardinality
>> issubs =: [ -: sand NB. Is x a subset of y?
>> pwrset =: a: ,~ [: /:~ ] (<@#~) 1 (,~"1) 2 (#"1~ {:)@#:@i...@^ #...@}:
>> NB. pwrset by Raul Miller, adapted
>> islist =: 1 = #...@$ NB. islist through isunique from validate.ijs
>> isboxed =: 0< L.
>> issorted =: -: /:~
>> isunique =: -: ~.
>> isset =: islist *. isboxed *. (a: -: {:) *. issorted@:}: *. isunique@:}:
>> NB. isset y asks, is array y a set?
>> iselement =:<@[ isin ] NB. Is array x an element of set y?
>>
>> NB. End
>>
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