The half-space is a powerful idea in these calculations.
Given a point in a plane (x, y) the points {(x', y') | x'>x ∧ y'>y} is a quarter plane of interest.
The same goes for eighth space in 3D.
Consider a brick whose edges are parallel to the axes.
Its volume is the eighth space of its least vertex, minus the eighth spaces of neighbors of that vertex, plus the eighth spaces of neighbors of those neighbors, etc.
We are not adding and subtracting sets of points, but their characteristic functions.