Generalized Inclusion-Exclusion

Sets are a foundational structure within mathematics and are commonly used as a building block for more complex structures. Just above this we have functions and sequences before an explosion of increasingly specialized structures. We propose a re-hanging of the tree with hybrid sets (that is, signed multi-sets), as well hybrid functions (functions with hybrid set domains) joining the ranks of sequences and functions. More than just an aesthetic change, this allows symbolic manipulation of structures in ways that might otherwise be cumbersome or inefficient. In particular, we will consider simplifying the product and sum of two piecewise functions or block matrices, integrating over hybrid set domains and the convolution of two piecewise interval functions