If U is {0,1} the subsets are
{} represented by 0 0
{1} represented by 0 1
(0} represented by 1 0
{0,1} represented by 1 1
but there is no way to represent a set like the power set (set of all subsets)
{ {}, {1}, {0}, {0,1} }
-- this set has four elements, and the largest set you can represent has two
elements. And it gets worse: you can't represent the power set of the power
set ...
On 10/10/2010 12:17 PM, Bo Jacoby wrote:
> If I understand this correctly, Kip's model only deals with finite sets which
> are all subsets of a finite set U, which can be represented by the numbers
> i.#U. Any subset of U can be represented by a bit vector, and bit vectors are
> manipulated by standard operations such as *. and +. and -. So I fail
> to see the purpose of Kip's representation of sets.
>
> --- Den søn 10/10/10 skrev Marshall Lochbaum<[email protected]>:
>
> Fra: Marshall Lochbaum<[email protected]>
> Emne: Re: [Jprogramming] Sets
> Til: "'Programming forum'"<[email protected]>
> Dato: søndag 10. oktober 2010 18.51
>
> 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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