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may mean one of two things. Most simply, it may mean the union of sets that are disjoint. But if two or more sets are not already disjoint, their disjoint union may be formed by modifying the sets to make them disjoint before forming the union of the modified sets. For instance two sets may be made disjoint by replacing each element by an ordered pair of the element and a binary value indicating whether it belongs to the first or second set.For families of more than two sets, one may similarly replace each element by an ordered pair of the element

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