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

*To*: mathgroup at smc.vnet.net*Subject*: [mg124294] Re: How to check whether an infinite set is closed under addition?*From*: Mobius ReX <aoirex at gmail.com>*Date*: Mon, 16 Jan 2012 17:07:41 -0500 (EST)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*References*: <201201150951.EAA19688@smc.vnet.net> <C9FCBB38-0E20-478B-97BC-BD57313E080F@mimuw.edu.pl>

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] &], >> 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] := > Complement[ > Select[Total[Subsets[Most[base], {2}], {2}], # <= Last[base] &], > 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>