The other way around is already implied by the previous property. Maybe it helps to consider finite sets. A finite set should clearly a minority, and the union

The other way around is already implied by the previous property. Maybe it helps to consider finite sets. A finite set should clearly a minority, and the union of two finite sets, though bigger, is still finite; similarly, the intersection of two co-finite sets is smaller, but still co-finite. Minority and majority are generalizations of finite and co-finite, respectively. (A set is co-finite if the set of elements in some universe that it doesn't contain is finite).