Question

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

Answer 1

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