graph theory – Is there a purely set-theoretic expression of the Euler characteristic?

The Euler characteristic of a graph is V – E + F. Where V is the number of vertices, E is the number of edges, and F is the number of faces.

Given a graph $X$, can we write $X$ as a set and find its Euler characteristic with a formula which only refers to cardinality and union?

I think we found a way, but is this interesting? We would like to find a way to write the Euler characteristic completely set-theoretically. And, we have found a way, but still working on it. Has this already been done?

Edit: the way to do it is to write the graph $X$ as a set of edges and faces. Then, let $A$ be the set of pairs of $X$, and $B$ be the set of triples of $X$. Then to compute the Euler Characteristic simply count
$$|cup A| + |B| – |A|.$$
This can generalize.