A fundamental inequality for the sum of sets of m-tuples

For n a positive integer and A₁, A₂, …, A_k sets of nonnegative integers, sufficient conditions are found which imply that the sum of the cardinalities of the sets $\text{{1, 2, …, n} ? A}{}_{\mathrm{i}}\text{(i = 1, 2, …, k)}$ does not exceed the cardinality of the intersection of {1, 2, …, n} and the number theoretic sum of the k sets. Some of the results are generalized to sets of m-tuples of nonnegative integers.