- #1

- 875

- 54

## Homework Statement

"Do there exist any event spaces with just six elements?"

## Homework Equations

## The Attempt at a Solution

Suppose ##F_1## is an event space with a non-trivial event ##A##. Then ##F_1=\{∅,A,A^c,Ω\}##. So ##inf(|F_1|) = 4##, since if you remove any of these events, ##F_1## is no longer closed under complements.

Suuppose ##F_2## is an event space with two non-trivial events ##A## and ##B##. Then ##F_2=\{∅,Ω,A,A^c,B,B^c,A\cup B, A^c\cap B^c,A\cap B, A^c\cup B^c,A^c\cap B, A\cup B^c, A\cap B^c, A^c\cup B\}##, and ##inf(|F_2|)=14## because if you remove any of these elements, then ##F_2## is no longer closed under set complements, intersections, and unions.

My argument is that there exists no integer ##i## between ##1## and ##2##, so there is no event space that has greater than four elements (##|F_1|##) and fewer than fourteen elements (##|F_2|##). I see a flaw, in that there might be event spaces with one event that have more than four elements, but I'm not totally sure that this is right...