# A question on two complement sets, which are subsets of a given set $X$

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 $$\{ x\in X: x\in A \wedge x\notin M\}$$?

• No. The definition of $M^{c}$ doesn't change because there are other subsets. What you have described is the intersection of $A$ and $M^c$. – John Douma Oct 31 '18 at 21:29
• @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 – D.Dave Oct 31 '18 at 21:33
• @JohnDouma Which is the same as $A\setminus M$. – Michael Hoppe Oct 31 '18 at 21:34
• $x$ cannot be in in both $A$ and $A^c$. – John Douma Oct 31 '18 at 21:35
• @fleablood I missunderstood it, now I understand it, I hope so :) – D.Dave Oct 31 '18 at 22:04

$$A$$ 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.