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

Answered by Joel0920 | 2024-06-10

The sets of numbers that are closed under addition include whole numbers (A), natural numbers (B), and integers (D). All these sets produce results that remain within the same set when added together. Negative integers (C) do not apply in the same context if incorporating a broader view of number properties.
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Answered by Joel0920 | 2024-12-26