# Produce a set with the following properties...

1. Mar 27, 2016

### jack476

1. The problem statement, all variables and given/known data
Produce an infinite collection of sets A1, A2, A3.... with the property that every Ai has an infinite number of elements, Ai∩Aj = ∅ for all i ≠ j, and the union of all Ai is equal to N.

2. Relevant equations
None provided.

3. The attempt at a solution
What I've come up with is that each Ai is the set of all natural numbers x with the properties that:
1.) The ith digit of x is i and
2.) x is not contained in Ai-1

It feels like it's right, but I don't know how to check, and the second criterion feels a bit cheap. Help?

2. Mar 27, 2016

### SammyS

Staff Emeritus
What numbers are in A13 for instance?

3. Mar 27, 2016

### jack476

Yeah, that wouldn't make much sense, would it. Whoops :/

4. Mar 27, 2016

### Dick

You actually have sort of the right idea, and step 2 isn't cheap. You are allowed to define sets recursively. You could, for example, think about prime numbers instead of digits. There are an infinite number of those, unlike digits. You were doing fine until you ran out of digits. And you could even stick with the digits if you redefine what you mean by the i'th digit.

Last edited: Mar 27, 2016
5. Mar 28, 2016

### HallsofIvy

Staff Emeritus
Looking at individual digits seems to me much too complicated. Just a condition of the values of x in $A_i$ should be enough. And saying "not in $A_{i-1}$ does not seem to me a good idea. Your definition of $A_i$ should make that automatic.

6. Mar 28, 2016

### Dick

Why? There is more than one way to skin a cat. I can think of several uncomplicated ways to do this using digits. And I don't see anything wrong with defining a set $A_i$ that depends on the definitions of the previous sets. I'll give jack476 an example. Define $A_1$ to be the set of all integers whose lowest order digit is NOT 1. That's an infinite set and it gives it gives you an infinite number of elements that are not in the set. That's makes it a good starting point. Now just keep winnowing it down making sure each set is infinite and you always have enough leftovers to build the next infinite set. And that they finally include all of the natural numbers. Here's another. Define $A_1$ to be all numbers that have no 1's in their decimal representation. Continue.

Last edited: Mar 28, 2016