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

• To: mathgroup at smc.vnet.net
• Subject: [mg124318] Re: How to check whether an infinite set is closed under addition?
• From: "Dr. Wolfgang Hintze" <weh at snafu.de>
• Date: Tue, 17 Jan 2012 03:19:16 -0500 (EST)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com
• References: <jeu7ma\$j85\$1@smc.vnet.net> <jf12qj\$5fb\$1@smc.vnet.net>

```Sorry, while I think the code is correct my choice for the sample data
was no good, of course.
Please vary the parameters in the line

> m = Union[Array[Random[Integer, {1, 5}] & , 5]]

from 5 to

m = Union[Array[Random[Integer, {1, a}] & , b]]

with a > b to generate non-trivial sets with gaps.

Best regards,
Wolfgang

"Dr. Wolfgang Hintze" <weh at snafu.de> schrieb im Newsbeitrag
news:jf12qj\$5fb\$1 at smc.vnet.net...
>
> "Rex" <aoirex at gmail.com> schrieb im Newsbeitrag
> news:jeu7ma\$j85\$1 at smc.vnet.net...
>> 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.
>>
>>
>
> You could proceed as follows:
>
> (* create some sample data for the set *)
> m = Union[Array[Random[Integer, {1, 5}] & , 5]]
> Out[44]=
> {1, 2, 3, 4, 5}
>
> (* Form all subsets with mor than 1 element *)
> ms = Subsets[m, {2, Length[m]}]
> Out[45]=
> {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4},
> {2, 5}, {3, 4}, {3, 5}, {4, 5}, {1, 2, 3},
> {1, 2, 4}, {1, 2, 5}, {1, 3, 4}, {1, 3, 5},
> {1, 4, 5}, {2, 3, 4}, {2, 3, 5}, {2, 4, 5},
> {3, 4, 5}, {1, 2, 3, 4}, {1, 2, 3, 5},
> {1, 2, 4, 5}, {1, 3, 4, 5}, {2, 3, 4, 5},
> {1, 2, 3, 4, 5}}
>
> (* compute all sums *)
> mp = (Plus @@ #1 & ) /@ ms
> Out[46]=
> {3, 4, 5, 6, 5, 6, 7, 7, 8, 9, 6, 7, 8, 8, 9, 10, 9,
> 10, 11, 12, 10, 11, 12, 13, 14, 15}
>
> (* find maximum of m in order to discard elements > maximum of m *)
> mx = Max[m]
> Out[47]=
> 5
>
> (* find all elements of m, the set of all sums, less than mx *)
> mp1 = Select[mp, #1 < mx & ]
> Out[48]=
> {3, 4}
>
> (* the intersection shows if the set m is closed under addition *)
> Intersection[mp1, m]
> Out[56]=
> {3, 4}
>
> (* as a result, thwe logical variable clsd indicates the closedness
> of
> the set *)
> clsd = mp1 == Intersection[mp1, m]
> Out[55]=
> True
>
> Regards,
> Wolfgang
>
>

```

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