令 \(A_1, ..., A_n\) 爲元素個數互不相等的n個正整數集且滿足對任意的\(i，j\)， \(A_i \notsubset A_j\)。 求它們幷集的元素個數的最小值。

Let \(A_1 ,\ldots,A_n\) be finite subsets of positive integers, such that no two of the \(A_i\) have same number of elements and \(A_i\) is not a subset of \(A_j\) for \(i\ne j\). What is the smallest possible number of elements in \(\cup_i A_i\)?