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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!

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  • $\begingroup$ What exactly are you trying to do? For some applications of this, the symmetric difference might do what you want. $\endgroup$ – Milo Brandt Jun 12 '16 at 20:14
  • $\begingroup$ I have "tests" that I can run on an item. Each test will tell me if the item belongs to a set of elements or not. So I can take item $i$, run test $1$ and that'll return my $x_1$. I can then run further tests to determine if $i$ is in another set and take the intersections of the results of these tests. The goal is to find what element $i$ actually is, in the least number of tests since the tests are expensive. However, I don't know of any notation that can represent what I'm trying to achieve. Does that help clarify, or just muddle the problem? $\endgroup$ – Jonathan Gawrych Jun 12 '16 at 20:36
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    $\begingroup$ I see. If it were me, I might do something like say $x_{j,+}$ is the set of $i$ that would pass test $j$ and say $x_{j,-}$ is the set that would fail and demand $x_{j,+}=x_{j,-}^C$. You could also treat every test as a function $f_j$ from the input set to $\{0,1\}$ and treat the preimage $f_j^{-1}(0)$ and $f_j^{-1}(1)$ - i.e. the inputs leading to either of these results. Neither of these is specifically does what you want, but I think they're reasonable ways to express the problem. $\endgroup$ – Milo Brandt Jun 12 '16 at 20:56
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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?

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  • $\begingroup$ That seems the most reasonable and least complicated. It took me way too long to even remember "set compliment" so I wanted to make sure I wasn't overlooking any standard notation. Especially if it might have lead me to more information. $\endgroup$ – Jonathan Gawrych Jun 12 '16 at 23:31
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Usually, the symbol backslash is used to denote relative complements, and if that's what you mean then what I mean is if $A$ and $B$ are sets we would write $A$ \ $B$={$x$:$x$ is in $A$ but is not in $B$}.

Never mind that's probably not what you mean.

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