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

• To: mathgroup at smc.vnet.net
• Subject: [mg124302] Re: How to check whether an infinite set is closed under addition?
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Mon, 16 Jan 2012 17:10:32 -0500 (EST)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com

```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.
>
>

Let'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

```

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