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We have a set $X$ and $M$, which is a subset of $X$. There is a unique complement of $M$ with respect to $X$: $$ M^c = X\setminus M =\{x\in X : x \notin M\} $$ But what if we would have another subset $A$ of $X$ and a complement of $A$: $$ A^c = X\setminus A=\{x\in X: x \notin A\} $$ Would the complement of $M$ now be $\{ x\in X: x\in A \wedge x\notin M\}$?

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    $\begingroup$ No. The definition of $M^{c}$ doesn't change because there are other subsets. What you have described is the intersection of $A$ and $M^c$. $\endgroup$ – John Douma Oct 31 '18 at 21:29
  • $\begingroup$ @JohnDouma Oh, I meant Mc = {x∈X: x∈A Λ x∈Ac Λ x∉M}. I have to see it visualized with two sets, then I wouldve understand it $\endgroup$ – D.Dave Oct 31 '18 at 21:33
  • $\begingroup$ @JohnDouma Which is the same as $A\setminus M$. $\endgroup$ – Michael Hoppe Oct 31 '18 at 21:34
  • $\begingroup$ $x$ cannot be in in both $A$ and $A^c$. $\endgroup$ – John Douma Oct 31 '18 at 21:35
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    $\begingroup$ @fleablood I missunderstood it, now I understand it, I hope so :) $\endgroup$ – D.Dave Oct 31 '18 at 22:04
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$A$ is not relevent to $M$ at all and need not be mentioned at all. They have nothing to do with each other.

Let $X = ${swans, snowmen, crows, coal, bluebirds, igloos, apartments, blueberries}

Let $A = $ things that are white = {swans, snowmen, igloos}. Then $A^c$ = things that are not white= {crows, coal, bluebirds, apartments, blueberries}.

Let $M = $ places you can live ={igloos, apartments}. Then $M^c$ = = things you can't live in ={swans, snowmen, crows, coal, bluebirds, blueberries}

Let $D = $ birds = {swans, crows, bluebirds}. Then $D^c = $things that are not birds = { snowmen, coal, igloos, apartments, blueberries}.

Figuring out $A^c$ is has nothing to do with $M$ or $D$ whatsoever.

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