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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