I have a metric coverage which for a given set is defined as the ratio of number of items in the given set that are also in the union of all the sets over the number of items in the union, with an added constraint that the union comprises of those items that exist in at least two sets.
For example, let $A = \{1,2,3\}$, $B = \{2,3,4\}$ and $C = \{4,5\}$. Then $U = \{2,3,4\}$ and
$$\operatorname{coverage}(A) = \frac{|\{2,3\}| }{ |\{2,3,4\}| } = 0.66,$$
and
$$\operatorname{coverage}(C) = \frac{|\{4\}| }{ |\{2,3,4\}| }= 0.33.$$
How can I represent the equation of coverage using set theory notations?
Let K be a collection of sets.
Define the weak union of K as
{ x : exists distinct A,B in K with x in A $\cap$ B }.
For A in K, the coverage of A is |A|/|weak union K|.