reachestimate of the union of facebook targeting specs

I’m looking for a simple way to get the reachestimate of a union of targeting specs.

It looks like the flexible spec ( can’t provide a union of different specs.
The flexible targeting spec allows:

1. “AND” different specs (and not “OR”).

2. “OR” different segments (interests, behaviors…) in one targeting spec.

The second option provides a union, which is larger then the real union of the two targeting specs. For example:

audience_1 = (interest_1 INTERSECTION behavior_1)

audience_2 = (interest_2 INTERSECTION behavior_2)

Then if I build a new spec with the two interests & two behaviors then it will give me:

(interest_1 UNION interest_2) UNION (behavior_1 UNION behavior_2)

which is a larger group than:

(interest_1 INTERSECTION behavior_1) UNION (interest_2 INTERSECTION behavior_2)

The first option provides an intersection between targeting specs, not the union, and in case I want the union of more than 2 audiences it becomes more complicated, it requires using the inclusion-exclusion principle ( which allow finding the union using intersections, but that’s not as simple as I hoped it to be, and requires many API calls to get sizes of all the subsets of audiences in all sizes.

Source: facebook

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