For instance:

- $ |A \cup B \cup C|=(|A|+|B|+|C|)-(|A \cap B| + |A \cap C| + |B \cap C|)+|A \cap B \cap C|$
- $\chi(X) = F- E + V = \sum_{i} (-1)^i \text{rank}(H_i(X)) =\sum_{i} (-1)^i \text{rank}(C_i(X))$
- Differential forms/Exterior algebra
- $\partial \sigma = \sum_{i=0}^n (-1)^i \sigma|[v_0, \dots,\hat{v_i}, \dots, v_n]$

Points (2), (3), and (4) all fall under "homology stuff" so I would expect there to be a consistent explanation of the signs in these cases. In particular, given my extremely rough understanding of differential forms and simplices being analogous I would expect there to be a precise explanation of the origin of the alternating-ness in these cases. I know you can attribute some of this to orientations and the symmetric group being present in both.

I see a nebulous relation between (1) and (3) in that differential forms and cardinality of finite sets are both measures of sorts. I might extend this to (2) after reading about the length of a potato, which in particular pointed out the relevance of the euler characteristic in making measurements.

There is also a relation between (1) and (2) in that the formula in (1) generalizes to: Given a finite set $X$ and finitely many subsets covering it, the cardinality of $X$ is the euler characteristic of the nerve of this cover. Note: The nerve is a simplicial set with one $n$-simplex for each $n$-fold intersection; I am defining the euler characteristic of a simplicial set to be the alternating sum $\sum_i (-1)^ic_i$ where $c_i$ is the number of non degenerate $n$-simplices.

This interpretation of (1) also fuels a connection to (4) because there are simplices in both.

The one general statement that I can make about all of the above examples, although this doesn't explain the alternating signs, is that the particular forms are often invariants under some sorts of transformations or representations. For instance (1) expresses the cardinality of a set in terms of any number of decompositions of that set into subsets; regarding (2), we know the euler characteristic is an invariant of homotopy; regarding (3), differential forms transform appropriately under diffeomorphisms; and regarding (4), the boundary is invariant under orientation preserving maps of the simplex.

**What I'm looking for is a conceptual unified explanation of the alternating-ness in all of the above examples.** Such an explanation would optimally expand on each of the points I made above. (I might also be looking for a better word that "alternating-ness", but thats another question.)