Formalizing sentences in propositional logic

Question

I'm studying for an introductory mathematical logic exam. Could you help me with formalizing the following conditions in propositional logic?

Divide the subsets of $\mathbb{N}$ in "small" and "large" sets such that:

1. Every subset $u \subseteq \mathbb{N}$ is either "small" or "large"
2. A set is "large" iff its complement is "small"
3. Subets of "small" sets are also "small"
4. The union of two "small" sets is also "small"
5. All finite sets are "small"

Hint: Let $X_u$ for every $u \subseteq \mathbb{N}$ be a variable such that $X_u$ is true, if $u$ is "large".

I have to formalize the conditions (1)-(5) in propositional logic, however I have difficulties with that. I have to formalize a set of formulas for each of the conditions.

Could you please give me a hint on that or an example?

So far I have the following:

Let $\tau:=\{X_u:u \subseteq \mathbb{N}\}$ be the set of the variables and $I$ be an interpretation $I:\tau \rightarrow \{0,1\}$. Then $X_L:=\{u \subseteq \mathbb{N}:I(X_u)=1\}$ is the set that contains all large subsets and $X_S:=\{u \subseteq \mathbb{N}:I(X_u)=0\}$ is the set that contains all small subsets.

Is this so far correct?

Answer 1

It is not clear exactly what this 'formalization' is supposed to look like; if you have to use propositional logic alone, I really don't see any way to do that. So, it must be the case that other mathematical symbols and notation is allowed (indeed, we need to talk about all kinds of set operations for sure!).

Now, if we had quantifiers, all this would be a piece of cake:

1. $\forall u \subseteq \mathbb{N}: (X_u \lor \neg X_u)$ (This is actually a tatutology!)

2. $\forall u \subseteq \mathbb{N}: (X_u \leftrightarrow \neg X_u^C)$

(if you need to define complement logically, you can use $\forall u \subseteq \mathbb{N} \ \forall n \in \mathbb{N} : (n \in X_u \leftrightarrow n \not \in X_u^C)$ and if you don't like the $X_u^C$ notation, you can use: $\forall u \subseteq \mathbb{N}: \forall v \subseteq \mathbb{N}: (\forall n \in \mathbb{N} (n \in X_u \leftrightarrow n \not \in X_v) \rightarrow (X_u \leftrightarrow \neg X_v))$

3. $\forall u \subseteq \mathbb{N}: (X_u \rightarrow \forall v \subseteq u: X_v)$

4. $\forall u \subseteq \mathbb{N}: \forall v \subseteq \mathbb{N}: ((X_u \land X_v) \rightarrow X_{u \cup v})$

(if you don't like to use $X_{u \cup v}$, you can do $\forall u \subseteq \mathbb{N}: \forall v \subseteq \mathbb{N}: \forall w \subseteq \mathbb{N}:(\forall n \in \mathbb{N} : (n \in w \leftrightarrow (n \in u \land n \in v)) \rightarrow ((X_u \land X_v) \rightarrow X_w))$

5. $\forall u \subseteq \mathbb{N}: \forall n \in \mathbb{N} : (|u| = n \rightarrow X_u)$

Without quantifiers though, I am really at a loss as to how to do this ...