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Let F be a finite nonempty family of finite nonempty sets. We prove thefollowing: (i) F satisfies the condition of the title if and only if for everypair of distinct subfamilies {A_1,...,A_r}, {B_1,...,B_s} of F, the union ofthe A_i is different from the union of the B_i. (ii) If F satisfies thecondition of the title, then the number of subsets of the union of the membersof F containing at least one set of F is odd. We give two applications of theseresults, one to number theory and one to commutative algebra.

Families of finite sets in which no set is covered by the union of the others