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