Complement for joint POVMs? 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.
I've come to the conclusion that the definition of the complement as I've written doesn't make sense for POVMs. It assumes that the product of two elements is a simultaneous event which is false. 

What I found is, because I'm dealing with probabilities and for me this was a "fail" event, I just switch my notion to 1 - "success" and I don't have to use any complements.