Two families of sets F and G are said to be cross s -union if for any F ∈ F and G ∈ G , | F ∪ G | ≤ s . In 2021, Frankl and Wong proved that if F , G ⊆ 2 [ n ] are non-empty cross s -union, then | F | + | G | ≤ ∑ i = 0 s n i + 1 . Moreover, for s < n − 1 , equality holds if and only if F , G = { 0̸ } , { G ⊆ [ n ] : | G | ≤ s } . In this paper, we give a new method to prove this result. Our method also allows us to establish a vector space version and a hereditary family extension. As a byproduct, we revisit the vector space version of the Katona s -union theorem due to Frankl and Tokushige, and characterize the extremal families for the case s = n − 1 .

There is a transition from a fuzzy set to crisp set. Therefore, we can obtain a conservative risk value, a venture risk value and a maximum probability risk value. Under such an alpha level, three risk values can be calculated. As alpha adopts all values throughout the set, it is possible to obtain a series of risk values. Therefore, the fuzzy risk may be a multi-valued risk or set-valued risk. Calculation of the fuzzy expected value of Yiwu city's water resource risk has been performed based on the interior-outer set model. Selection of an alpha value depends on the confidence in different groups of people, while selection of a conservative risk value or venture risk value depends on the risk preference of these people.