I'm trying to relate some notions of set theory to POVMs. I firstly explain the scenario with set theory and then in the POVM setting.
For some finite $N \in \mathbb{N}$, let $A_i$ and $B_i$ for $i=1,...,N$ be sets and assume there is some larger set $C$ such that $A_i\subset C, B_i\subset C$ for all $i$. Let's say for $J=\{1,...,N/2\}$ (just take $N$ even), I'd like to compute $$ \left(\bigcup_{i\in J} A_i\times B_i\right)^c.$$
By De Morgan's laws we can write, $$\left(\bigcup_{i\in J} A_i\times B_i \right)^c = \bigcap_{i\in J} (A_i\times B_i)^c = \bigcap_{i\in J} A_i^c\times B_i \cup A_i\times B_i^c \cup A_i^c \times B_i^c$$
Now I'd like to do something similar for POVMs. Let's say I have two POVMs, $A = \{A_1,...,A_N \}$ and $B=\{B_1,...,B_N\}$.
For either POVM, I think it's clear how to define a complement for a subset. For example, if I want the complement of $\{A_2, A_3\}$, then I can do $\mathbb{1} - A_2 - A_3$. My interpretation is, $\mathbb{1} - A_2 - A_3$ encompasses all events except $A_2$ or $A_3$. Now I want to define a (or be given a known) complement for a joint measurement.
Let's again say I have some $J = \{1, ..., N/2\}$ and I want to determine the complement of the sum joint measurements indexed over $J$. That is, I want somehow to define
$$\left(\sum_{i\in J}A_i \otimes B_i\right)^c$$
My first thought was to have
$$\left(\sum_{i\in J}A_i \otimes B_i\right)^c = \mathbb{1} - \sum_{i\in J}A_i \otimes B_i$$
But this only encompasses the analog of $\bigcup_{i \in J} A_i^c \times B_i^c$ and misses the analogs of $A_i^c \times B_i$ and $A_i\times B_i^c$.
Then I thought, in a De Morgan sense
$$\left(\sum_{i\in J}A_i \otimes B_i\right)^c = \prod_{i\in J} (A_i \otimes B_i)^c = \prod_{i\in J} (\mathbb{1} - A_i)\otimes B_i + A_i \otimes (\mathbb{1}-B_i) + (\mathbb{1} - A_i)\otimes (\mathbb{1}-B_i)$$
This I think is logical, but I've not yet convinced myself that it's true. Has anyone had a similar scenario?
Thanks for your time and sorry if the question isn't clear.
