# Finite and Countable union of countable sets

1. Mar 21, 2012

### 6.28318531

1. The problem statement, all variables and given/known data
Show the following sets are countable;
i) A finite union of countable sets.
ii) A countable union of countable sets.

2. Relevant equations

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

3. The attempt at a solution
Part i) Well I suppose you could start by considering V1,V2,......Vn 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?

2. Mar 21, 2012

### micromass

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

3. Mar 21, 2012

### 6.28318531

So something like V =$\bigcup$Vij, then arrange the elements Vij 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?

4. Mar 21, 2012

### micromass

Staff Emeritus
What are the $V_{ij}$?

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