# Thread: Qestion in set theory

1. ## 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 :-)

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