Partition of Sets

[Seeker555]

$\displaystyle A_i\subset \Omega$ 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.

 

[pka]

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 $\displaystyle A_n=(n-1,n]$.

$\displaystyle A_i\subset \Omega$ 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.

I don't understand the above notation.

 

[Seeker555]

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]

ignore the dollar sign, that's just a mistake.

 

[pka]

the rest of the notation is as it is written by the university professors.

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.

 

[Seeker555]

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.

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.