# Qestion in set theory

• Dec 17th 2008, 03:26 AM
John G
Qestion in set theory
L is the set of cofinite subsets of N (i'm not sure if i understand the concept of cofinite sets).
I need to prove that L is countable. How do i prove this? What to do?

Thanks for any kind of help :-)
• Dec 17th 2008, 01:28 PM
Plato
Quote:

Originally Posted by John G
L is the set of cofinite subsets of N (i'm not sure if i understand the concept of cofinite sets). I need to prove that L is countable.

I will assume that $\mathbb{N} = \mathbb{Z}^ +$ the positive integers.
A cofinite set has the property that its complement is finite.
Thus each finite subset of $\mathbb{N}$ corresponds to a cofinite subset of $\mathbb{N}$.
The collection of finite subsets of $\mathbb{N}$ is a countable set.
Lets say that $\mathbb{L}$ is collections of finite sets, define a mapping from $\mathbb{L}$ to $\mathbb{N}$ as follows.
Make a listing of the prime numbers, $\left\{ {p_1 ,p_2 ,p_3 ,p_4 , \cdots } \right\}$.
$H: \mathbb{L} \mapsto \mathbb{N}$ by
$H(B) = \left\{ {\begin{array}{*{20}c}
{1,} & {B = \emptyset } \\
{\prod\limits_{x \in B} {p_x } } & {else} \\
\end{array} } \right.$

Then show that $H$ is injective.