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Suppose there exists a collection of finite sets which is finite.

We pick up the minimal sub-collection such that any set in the collection can be expressed as a union of sets in the sub-collection.

Is the minimal sub-collection unique?

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set-theory tag is not appropriate. Maybe elementary-set-theory is. – Aryabhata May 14 '11 at 16:55

Yes it is unique. Let us consider two minimal sub-collections $\mathcal A, \mathcal B$. Neither contains the other by their minimality. Since everything is finite, let $A\in \mathcal A\setminus\mathcal B$ be an element of minimal cardinality. Now $A$ can be expressed as a union of element of $\mathcal B$, which all need to be of smaller cardinality than $A$ (or same but $A\not\in\mathcal B$), but $\mathcal A$ then contains all of them, letting $A$ be expressed by a union of elements of $\mathcal A$ contradicting the minimality of $\mathcal A$.

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