Produce a set with the following properties....

[jack476]

Homework Statement

Produce an infinite collection of sets A₁, A₂, A₃... with the property that every A_i has an infinite number of elements, A_i∩A_j = ∅ for all i ≠ j, and the union of all A_i is equal to N.

Homework Equations

None provided.

The Attempt at a Solution

What I've come up with is that each A_i is the set of all natural numbers x with the properties that:
1.) The i^th digit of x is i and
2.) x is not contained in A_i-1

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

 

[SammyS]

Homework Statement

Produce an infinite collection of sets A₁, A₂, A₃... with the property that every A_i has an infinite number of elements, A_i∩A_j = ∅ for all i ≠ j, and the union of all A_i is equal to N.

Homework Equations

None provided.

The Attempt at a Solution

What I've come up with is that each A_i is the set of all natural numbers x with the properties that:
1.) The i^th digit of x is i and
2.) x is not contained in A_i-1

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

What numbers are in A₁₃ for instance?

 

[jack476]

What numbers are in A₁₃ for instance?

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

 

[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.

 

[HallsofIvy]

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.

 

[Dick]

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.

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.