# Set Theory Question

1. Mar 18, 2015

### AlephNumbers

Lately I have been attempting (and failing miserably at) whatever sample Putnam questions I can find on the internet. Here is my latest endeavor. I found this question on the Kansas State University website, so I think I am allowed to post it. I must warn you that I know almost nothing about set theory.

Can a countably infinite set have an uncountable collection of non-empty subsets such that the intersection of any two of them is finite?

I know that the set of all integers is "countably" infinite.
Let I be the set containing all integers. The set I contains an infinite number of subsets each containing any number of the infinite number of elements in the set I, so the number of subsets of I is uncountable. However, any two subsets of I can be shown to have a finite intersection.
For example G={1,2,3,4} , F={3,4,5}
G and F have 2 elements in common; the elements 3 and 4 belong to both sets

I don't know if this would be a sufficient answer. It's awfully wordy and does not contain much mathematical rigor.

2. Mar 18, 2015

### Staff: Mentor

What about A={2,4,6,8,...} and B={3,6,9,12,...} with their intersection {6,12,18,...}?
You need sets with infinite members, otherwise you do not get an uncountable number of subsets.

I like that question. It has a nice answer.

Last edited: Mar 18, 2015
3. Mar 18, 2015

### Dick

It doesn't contain any rigor. And tackling Putnam questions while saying you know almost nothing about set theory is a bold move. I agree with mfb that it is a nice question and does have an easy answer. Here's a hint. You can split $I$ into three disjoint sets where two contain a countably infinite number of members and one has a finite number. Show that and use it.

Last edited: Mar 18, 2015
4. Mar 19, 2015

### AlephNumbers

Hmm. I think I'll put this one on the backburner. Thank you for the input.