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 :-)

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- Dec 17th 2008, 02:26 AMJohn GQestion 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, 12:28 PMPlato

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.