Power set of N

**Pythagorean12** · December 2nd 2009, 06:01 PM

Is there an uncountable family

such that

∩

is finite for all distinct

,

∈

?
(here

denotes a power set for N).

**Drexel28** · December 2nd 2009, 08:56 PM

Originally Posted by Pythagorean12

Is there an uncountable family

such that

∩

is finite for all distinct

,

∈

?
(here

denotes a power set for N).

Before you even try to construct an example or craft a proof, what do you think the answer is?

**eurialo** · December 4th 2009, 07:35 AM

I don't know if it can be useful in your case, but there is a general theorem that states that, given a cardinal $\kappa \geq \omega$ such that $2^{< \kappa} = \kappa$ , then there exists an almost disjoint family $F \subset P(\kappa)$ with $|F| = 2^\kappa$ .
This theorem, with $\kappa = \omega$ answers your question.

**sidor** · December 7th 2009, 06:16 AM

Come on, do not take a sledgehammer to crack a nut. There is a simple example:

Let the $P_i$ be such infinite set that $p_i^k \in P_i$ , where $p_i$ is the i-th prime number and $k \in \mathbb{N}$ .

Then infinite family $\{ P_1, P_2, P_3, ... \}$ is an example satisfying the requirements. Because for every distinct $P_i \cap P_j = \{1\}$ and $P_i \subseteq \mathcal{P}(\mathbb{N})$ and there are infinitely many primes.

**tonio** · December 7th 2009, 06:28 AM

Originally Posted by sidor

Come on, do not take a sledgehammer to crack a nut. There is a simple example:

Let the $P_i$ be such infinite set that $p_i^k \in P_i$ , where $p_i$ is the i-th prime number and $k \in \mathbb{N}$ .

Then infinite family $\{ P_1, P_2, P_3, ... \}$ is an example satisfying the requirements. Because for every distinct $P_i \cap P_j = \{1\}$ and $P_i \subseteq \mathcal{P}(\mathbb{N})$ and there are infinitely many primes.

The family $\{P_1,\ldots\}\subset P(\mathbb{N})$ is not uncountable, as requested in the OP.

Tonio

**Shanks** · December 7th 2009, 06:37 PM

I read a problem like this somewhere.
since there is a bijective mapping between N and Q, let A be the collection of all rational number cauchy sequence converging to 0, B be the collection of all rational number cauchy sequence converging to 1.
A union B is a uncountable set. for each a in A, and b in B, there are only finite common elements.

**Drexel28** · December 7th 2009, 09:26 PM

Originally Posted by Shanks

I read a problem like this somewhere.
since there is a bijective mapping between N and Q, let A be the collection of all rational number cauchy sequence converging to 0, B be the collection of all rational number cauchy sequence converging to 1.
A union B is a uncountable set. for each a in A, and b in B, there are only finite common elements.

I actually did a similar problem to this (as you described). The question was "Does there exists an uncountable family of subsets of a countable set such that the interesction of any two is finite?" The answer is yes. I gave two examples. I actually did it on the forum if you are interested. But! The OP said that the sets need to be subsets of $\mathbb{N}$ and as your sets are not this doesn't work. I refuse to actually answer the question upon the grounds that the OP has NEVER shown any effort.

**Shanks** · December 14th 2009, 08:17 PM

That is Why I said there is a bijective mapping between N and Q.
This bijective mapping play a role in this example.

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