[Math] Notation for conditional set complement

Question

As far as I know, given $U=\{1,2,3,4,5,6\},A=\{1,2,3\}$ the notation for its set complement is $A^C = \{4,5,6\}$

Is there any sort of notation for a conditional set complement? For example, lets say I had a true/false variable $x_1$ who determines in an equation if $A$ should be itself or its complement. I think I could do a piece-wise function like so:

$$B=\begin{cases}
A& \text{if $x_1$ is true},\\
A^C& \text{if $x_1$ is false}.
\end{cases}$$

but I actually have many conditionally complemented sets that I am using. If I use a single piece-wise function, that would be $2^{n}$ cases, or I could use set operators between $n$ different piece-wise functions, but that seems very verbose. Thanks!

Answer 1

Could you say:

For $i=1,\dots,n$ let

$$B_i=\begin{cases} A_i& \text{if $x_i$ is true,}\\ A_i^C& \text{if $x_i$ is false.} \end{cases}$$

Now let $B=B_1\cap\dots\cap B_n$.

Or something to that effect?