Finite and Countable union of countable sets

[6.28318531]

Homework Statement

Show the following sets are countable;
i) A finite union of countable sets.
ii) A countable union of countable sets.

Homework Equations

A set X, is countable if there exists a bijection f: X → Z

The Attempt at a Solution

Part i) Well I suppose you could start by considering V₁,V₂,...V_n countable sets. Let V = $\bigcup$$^{n}_{i=1}$V$_{n}$, and then we have to define some general bijection between Z and V?

Part ii) Is there a way to write out all the elements of a collection of sets as a grid, similar to showing why the rational numbers are countable, and then move through them in some ordered manner, so that we can create a bijection? Is there a way to formalise this?

 

[micromass]

Part ii) Is there a way to write out all the elements of a collection of sets as a grid, similar to showing why the rational numbers are countable, and then move through them in some ordered manner, so that we can create a bijection? Is there a way to formalise this?

Yes. The proof is very similar to showing that the rationals are countable. Try something like that.

 

[6.28318531]

So something like V =$\bigcup$V_ij, then arrange the elements V_ij in a grid (like a matrix )
then choose V₁₁,V₂₁,_V12,V₃₁...etc, so then you can simply map

f V → Z, with f(n) = n the nth element of the list?

 

[micromass]

So something like V =$\bigcup$V_ij, then arrange the elements V_ij in a grid (like a matrix )

What are the $V_{ij}$?