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Assume that $\{A_{i}\}_{i\in I}$ is a collection of non-empty sets indexed by an arbitrary indexing set $I$. How can I construct a pairwise disjoint collection $\{A'_{i}\}_{i\in I}$ such that $A'_{i}\subset A_{i}$ for every $i$, and $${\bigcup}_{i\in I}A_{i}'={\bigcup}_{i\in I}A_{i}?$$ (By pairwise disjoint, I mean, of course, that $i\neq j$ implies $A_{i}'\cap A_{j}' = \emptyset.$) If $I$ were finite, I could use the inclusion-exclusion principle. But would this work when $I$ is countably infinite, or when $I$ is uncountable?

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Assuming $I$ is truly arbitrary, I think you would need choice. Well order $I$ with order type $\alpha$, and define sets $A'_\beta$, $\beta < \alpha$ by $$A'_\beta = A_\beta \setminus \bigcup_{\gamma<\beta} A_\gamma$$

The $A_\beta$ are pairwise disjoint by construction, and their union is still the same: for every $x \in \bigcup A_i$, there is unique $\beta$ with $x \in A'_\beta$; in fact, $\beta = \min \{ \gamma : x \in A_\gamma \}$.

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