Which sets of numbers are closed under addition?
choose all answers that are correct.
a.
whole numbers
b.
natural numbers
c.
negative integers
d.
integers

The natural numbers are well-ordered: which means every set of natural numbers has a least element. 

So suppose S is a set of natural numbers closed under addition. 

Let k be the smallest element of S. 

Then S contains:

k,k+k, k+k+k,etc

in other words S must contain all multiples of k.

could S contain other elements besides multiples of k?

suppose it did. suppose it contained m.

then we get all natural numbers of the form ak + bm.

for example, if k = 2, m = 3, S might be:

S = {2,3,4,5,6,7,8,} = N - {0,1}.

note we can write this set as:

{2 + k(gcd(2,3)): k in N}

this can be generalized to more than a pair of numbers

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