Inequalities for cross-unions of collections of finite sets

Some inequalities for cross-unions of families of finite sets are proved that are related to the problem of minimizing the union-closure of a uniform family of given size. The cross-union of two families F and G of subsets of [n]={1,2,…,n}[n]={1,2,…,n} is the family F∨G={F∪G:F∈F,G∈G}. It is shown that |F∨G|/|F|≥|G∨B_n|/2ⁿ|, where B_n denotes the power set of [n]. Besides, the problem of minimizing |F∨G|over all union-closed FF and G∨G generated by a given number r of singletons and a given number s > (r 2)  of two-sets, respectively, is solved.