Constructing partitions of $\mathcal P(\Bbb R)$ under certain constraints.

Question

I tried solving the following subtasks. I would like to verify my answers and, if valid, justify them further. I thought it would be better to clarify all of my possible misconceptions in one post as the subtasks may be closely related to each other.

Write down a partition $\{A_i\mid i\in I\}$ of the set $\mathcal P(\Bbb R)$ so that:

1. $A_i$ is finite, $\forall i\in I$
2. $A_i$ is countable, $\forall i\in I$
3. $\operatorname{card} A_i=2^c,\forall i\in I$ and $\operatorname{card}I=c.$
4. $\operatorname{card} A_i=2^c,\forall i\in I$ and $\operatorname{card}I=2^c.$

My attempt:

1. I just took $A_i=\{B_i,\Bbb R\setminus B_i\},i\in I.$

2. I wanted to use the following result (proven in my materials):

Let $S:=\{A\subseteq\Bbb N\mid A\text{ is finite }\}.$ Then $\operatorname{card} S=\aleph_0.$

I think the same should hold if we replace $\Bbb N$ by $\Bbb Z.\quad$ Let $\color{red}{S\in\mathcal P(\Bbb R\setminus\Bbb Z)}.$

I think sets like $A_{\color{red}S}=\{\color{red}S\cup A\mid A\subseteq Z\text{ and } A\text{ is finite }\}$ could form the desired partition.

3. Since $[-1,1]\sim\Bbb R,\space\boxed{\mathcal P([-1,1])=2^c}.\quad$ Let $x>1$ be arbitrary. We have $$\mathcal P([-x,x]\setminus[-1,1])\setminus\{\emptyset\}\subseteq\mathcal P([-x,x])\setminus\mathcal P([-1,1])\subseteq\mathcal P([-x,x])$$ and $$\operatorname{card}\mathcal P([-x,x]\setminus[-1,1])\setminus\{\emptyset\}=\operatorname{card}\mathcal P([-x,x])=2^c,$$ therefore $\operatorname{card}\mathcal P([-x,x])\setminus\mathcal P([-1,1])=2^c.$

So, we could consider sets of the form $$A_y=\mathcal P([-y,y])\setminus\mathcal P([-x,x]),x<y$$ for the desired partition.

4. Let $X\subset(-\infty,0)$ be arbitrary and let's consider $\mathcal P([0,+\infty)).$ I took $A_X:=\{X\cup Y\mid Y\in\mathcal P([0,+\infty))\}$ because $\operatorname{card}A_X=2^c$ for each of the total $2^c$ sets $X\in\mathcal P((-\infty,0)).$

Is there any fallacy and how can I improve my reasoning? Are there any simpler examples?

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EDIT:

I think my answer to the second task is wrong as I forgot the infinite subsets of $\Bbb Z,$ but I can't modify what I've done so far in order to fix that.

Answer 1

You have done pretty well. Your construction for the third and last subtasks are clear and simple.

Let us check the first subtask. Your construction did not mention what is the index set $I$ nor what is the set $B_i$ that corresponds to $i$. It looks you meant that $I$ is the quotient space of $\mathcal P(\Bbb R)$ under the equivalence relation such that $S\sim \Bbb R\setminus S$ for all $S\subseteq\Bbb R$. Then, of course, your construction is correct.

A simpler construction is $A_i=\{i\}$ for $i\in I=\mathcal P(\Bbb R)$.

On the second subtask, you are near to a solution.

Let us assume that "countable" means "countably infinite". Otherwise if a finite set is also countable, a construction for the first subtask would do.

Define a homogeneous binary relation $\sim_f$ over $\mathcal P(\Bbb R)$ so that $S\sim_fT$ iff the symmetric difference between $S$ and $T$ is a finite subset of $\Bbb N$. That is, one subset of $\Bbb R$ is related to another iff they differ by finitely many natural numbers. It is straightforward to check $\sim_f$ is an equivalence relation. This equivalence relation partitions $\mathcal P(\Bbb R)$ into equivalence classes. We can check that each equivalence class is a countable set.

The solution above to the second subtask generalizes immediately to the power set of any infinite set.