Partition of Sets

S

Seeker555

New member
Joined
Jan 14, 2012
Messages
25
\(\displaystyle A_i\subset \Omega\) Let \(\displaystyle \Omega\) be any set
\(\displaystyle i\in I,\ I=\mathbb{N}\)

\(\displaystyle \mathcal{P} =\ \{A_i\ |\ i\in I\}\)
is a partition of \(\displaystyle \Omega\) if each set \(\displaystyle A_i\) is not empty. The sets \(\displaystyle A_i\) are mutually disjoint.

Now let \(\displaystyle \Omega =\ \mathbb{R}\). Give an example of a partition of \(\displaystyle \mathbb{R}\) such that a) I={1,2}, b) I=(1,2,3), c) I=(\(\displaystyle \mathbb{N}\).

Now given the above,
\(\displaystyle \mathcal{F}\ =\ {\bigcup}{i\in J}\ A_i\ |J\subset I\) (here the or than symbol is mean to have \(\displaystyle i\in J\) underneath it sorry)

Write down the collection \(\displaystyle \mathcal{F}\\) explicitly for the examples of P you gave in
part (i) for I = (1,2) and I=(1,2,3). And prove that \(\displaystyle \mathcal{F}\\) is a field.

So any help much appreciated and i'm sure if you only give an answer for just one of a), b) i'll be able to work out the others. Thanks.
 
Seeker555 said:
Now let \(\displaystyle \Omega =\ \mathbb{R}\). Give an example of a partition of \(\displaystyle \mathbb{R}\) such that a) I={1,2}, b) I=(1,2,3), c) I=(\(\displaystyle \mathbb{N}\).
For part c). Let \(\displaystyle A_0=(-\infty,0]\), if \(\displaystyle n\in\mathbb{Z}^+\) define \(\displaystyle A_n=(n-1,n]\).
Seeker555 said:
\(\displaystyle A_i\subset \Omega\) Let \(\displaystyle \Omega\) be any set \(\displaystyle i\in I,\ I=\mathbb{N}\)

\(\displaystyle \mathcal{F}\ =\ {\bigcup}{i\in J}\ A_i\ |J\subset I\) (here the or than symbol is mean to have \(\displaystyle i\in J\) underneath it sorry)
Write down the collection \(\displaystyle \mathcal{F}\\) explicitly for the examples of P you gave in
part (i) for I = (1,2) and I=(1,2,3). And prove that \(\displaystyle \mathcal{F}\\) is a field.
Click to expand...
I don't understand the above notation.
Click to expand...
 
The big cup is meant to have i elemnt of J underneath it. the rest of the notation is as it is written by the university professors.
 
Seeker555 said:
the rest of the notation is as it is written by the university professors.
Click to expand...
I am a retired university professor who taught set theory for thirty years and I still have no idea what \(\displaystyle F = \left {\bigcup\limits_{i \in J} {A_i } } \right|\,J \subset I\) means.
 
Last edited:
pka said:
I am a retired university professor who taught set theory for thirty years and I still have no idea what \(\displaystyle F = \left {\bigcup\limits_{i \in J} {A_i } } \right|\,J \subset I\) means.
Click to expand...

hehe ****. though i should say the F is written in a styalised way. (but i can't imagine that would make much of a difference). **** no wonder i dont have a clue what to do.

i couldn't ask, what latex did you use to get the i element of j underneath the big cup? that would be a great help.
 
You must log in or register to reply here.
Top