Split the natural numbers into three sets A, B, and C so that the sets have nothing in common, they each are countably infinite, and A U B U C = the set of natural numbers.

So here's what I got
A: 2,4,6,8,10...
B:1,3,5,7,9...
C: ?

I don't know what to do for C because it cannot contain any of the number in A or B. Do I just pick a number and have in continually repeat?

how about:

powers of 2
powers of 3
everything else

or, pick numbers with different remainders when divided by 3:

3n+0: 3,6,9,12,...
3n+1: 4,7,10,13,...
3n+2: 5,8,11,14,...

To construct set C, you can choose a different pattern or rule that is distinct from both sets A and B. One way to do this is by using the concept of prime numbers.

Set A: Contains all even numbers starting from 2, i.e., A = {2, 4, 6, 8, 10, ...}.
Set B: Contains all odd numbers starting from 1, i.e., B = {1, 3, 5, 7, 9, ...}.

For set C, we can use prime numbers. Recall that prime numbers are numbers greater than 1 that are divisible only by 1 and themselves. Here's one way to construct set C using prime numbers:

Set C: Contains all prime numbers, i.e., C = {2, 3, 5, 7, 11, 13, 17, 19, 23, ...}.

Since prime numbers are distinct from both even and odd numbers, this ensures that sets A, B, and C are all disjoint (they have nothing in common) and each set is countably infinite.

By constructing the sets in this way, you ensure that every natural number is included in the union of sets A, B, and C, as A U B U C equals the set of natural numbers.