Re: How to check whether an infinite set is closed under addition?

*To*: mathgroup at smc.vnet.net*Subject*: [mg124296] Re: How to check whether an infinite set is closed under addition?*From*: Mobius ReX <aoirex at gmail.com>*Date*: Mon, 16 Jan 2012 17:08:24 -0500 (EST)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*References*: <201201150951.EAA19688@smc.vnet.net> <C9FCBB38-0E20-478B-97BC-BD57313E080F@mimuw.edu.pl>

Sorry the condition of a set A to be closed under addition should be `a_i*n + a_j*m =E2=88=88 A` for any positive integers `n` and `m` for any a_i =E2=88=88 A. On Sun, Jan 15, 2012 at 4:55 PM, Mobius ReX <aoirex at gmail.com> wrote: > I just realized that there should be a further story here. > To check whether the set is complete, we should also check whether the > addition of a_i with itself with multiple times is also included in > the set. > > This means that for a set A to be closed, its elements a_i =E2=88=88 A should satisfy > > 1). a_i + a_j =E2=88=88 A for any a_i and a_j; > 2). a_i * n =E2=88=88 =C2 A for any positive integer n. > > Any more tips to quickly check the second condition? > > Thanks. > > Best, > Rex > > > > On Sun, Jan 15, 2012 at 2:57 PM, Mobius ReX <aoirex at gmail.com> wrote: >> Dear Andrzej, >> >> Cool! Thanks a lot. >> >> Best, >> Renjun >> >> >> On Sun, Jan 15, 2012 at 4:22 AM, Andrzej Kozlowski <akoz at mimuw.edu.pl> wrote: >>> >>> On 15 Jan 2012, at 12:00, Andrzej Kozlowski wrote: >>> >>>> Let >>>> On 15 Jan 2012, at 10:51, Rex wrote: >>>> >>>>> Given k positive numbers a_1<a_2<a_3<...<a_k, and all integers greater >>>>> than a_k, we want to check whether this set {a_1, a_2, a_3,...a_k, a_k >>>>> + 1, a_k+2, ......} is closed under addition. >>>>> >>>>> Is there any easy way to do this? any functions that we could use in >>>>> Mathematica? >>>>> >>>>> Your help will be greatly appreciated. >>>>> >>>>> >>>> >>>> Lest's call your set {a1,a2,...,a3} "base". Then: >>>> >>>> closedQ[base_List] := >>>> Complement[Select[Total[Subsets[base, {2}], {2}], # <= Max[base] &], >>>> =C2 base] == {} >>>> >>>> For example: >>>> >>>> closedQ[{1, 2, 3}] >>>> >>>> True >>>> >>>> closedQ[{1, 4, 6, 7}] >>>> >>>> False >>>> >>>> Andrzej Kozlowski >>>> >>> >>> The program above does dome unnecessary comparisons since anything added to the largest element of the base will obviously included in the complete set. If the base is ordered, a better program will be: >>> >>> closedQ[base_List] := >>> =C2 Complement[ >>> =C2 Select[Total[Subsets[Most[base], {2}], {2}], # <= Last[base] &], >>> =C2 base] == {} >>> >>> If it is not ordered we could use Sort[base] in place of base. >>> >>> Andrzej Kozlowski

**References**:**How to check whether an infinite set is closed under addition?***From:*Rex <aoirex@gmail.com>