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Let $C$ and $D$ be subsets of $X$. Prove that $$(X\setminus C)\cap D=D\setminus C$$ My workings: Showing two sets are equal amounts to showing inclusion both ways so I want to show $$(X\setminus C)\cap D\subseteq C$$ and $$(X\setminus C)\cap D\supseteq D\setminus C$$ For the first one I did the following. Let $x\in (X\setminus C)\cap D$, then $x\in(X\setminus C)$ and $x\in D$. This implies that $x\in X$, $x\notin C$ and $x\in D$. Because $D\subseteq X$ this is equivalent to saying $x\in D$, $x\notin C$. Therefore $x\in D\setminus C$. this shows that $(X\setminus C)\cap D\subseteq D\setminus C$.

For the second part I ran into some trouble. This time let $x\in D\setminus C$. Then I thought since $D\subseteq X$ this implies $D\cap X=D$ and we can say $x\in (X\cap D)\setminus C$. I just don't know if this is completely correct to say, and it also doesn't finish the argument.

If anyone could help me out with this last part I would be grateful. Also it is very possible that the arguments in the first are not rigorous enough, if so please feel free to point out where I could improve. Thanks

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4 Answers 4

up vote 2 down vote accepted

Your first inclusion looks fine.

For the second, your thought pattern is fine; just need to put it in notation...we simply "commute" the "anded" statements.

$ \begin{align} x \in D \setminus C &\implies x \in D\;\text{ and}\;x \notin C \\ \\ &\implies x \in D \cap X = D\;\text{ and}\;x \notin C \\ \\ & \implies x \in D \text{ and}\; x \in X\;\text{ and}\;x \notin C \\ \\ &\implies x \in X\;\text{ and}\;x\notin C\;\text{ and}\;x \in D \\ \\ & \implies x \in (X\setminus C)\;\text{ and}\;x \in D \\ \\ & \implies x \in (X\setminus C)\cap D \end{align} $

So we've shown that $x \in D\setminus C \implies x\in (X\setminus C)\cap D$

Hence, $\;D\setminus C \subseteq (X\setminus C)\cap D$

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Thanks, so if I were to continue the argument I was working on would the reasoning $x\in (X\cap D)\setminus C$ implies $x\in (X\setminus C)\cap D$ be correct as well because $C,D\subseteq X$? –  Slugger Feb 11 '13 at 16:55
    
Absolutely, because we have "anded" statements x in X and x in D and x not in C...and commute them, validly, getting x in X and x not in C, and x in D. –  amWhy Feb 11 '13 at 17:00
    
Does this make sense now? X is the "superset" containing subsets C and D, as you argued... –  amWhy Feb 11 '13 at 17:05
    
Yes this makes a lot of sense thanks! I have a lot more of these questions I need to do so I might post one or two more if I am having trouble (or is it considered bad manners to upload several similar questions?) –  Slugger Feb 11 '13 at 17:10
    
What do you think of calling Superset, a Mother set? + –  Babak S. Feb 11 '13 at 18:15

$X \setminus C = C^C$.

$X \setminus C \cap D = C^C \cap D = D \setminus C$.

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Hey thanks for the answer, it is definitely a very nice argument because it is so short and concise. However, my teacher asked us to show inclusion both ways so I am afraid this will not be accepted. Thanks for the answer though, it is nice! –  Slugger Feb 11 '13 at 16:53
    
You can repeat the above twice, once with $=$ replaced by $\subset$ and another time with $=$ replaced by $\supset$. –  copper.hat Feb 11 '13 at 16:56

The first direction is well-handled. For the other direction, essentially the same argument works.

You want to prove if $x\in D\setminus C$, then $x$ is in $(X\setminus C)\cap D$. This is immediate, since $x$ is in $D$, in $X$ of course because $D$ is a subset of $X$, but not in $C$.

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Hey thanks for the feedback. I am not sure if I can see the typo, in the second part I am assuming that $x\in D\setminus C$ and want to show it is therefore in $(X\setminus C)\cap D$. Now because $D=X\cap D$ I argued that $D\setminus C = (X\cap D)\setminus C$. Maybe this is the typo you are referring to, if so, it is actually what I meant. –  Slugger Feb 11 '13 at 16:51
    
I guess I was surprised, since a direct attack works well, and is essentially the same as the forward direction that you handled with no trouble. –  André Nicolas Feb 11 '13 at 16:56
    
Yeah I can understand, this is my first encounter with proving these kinds of things. I find it difficult because the whole statement looks so obvious and a Venn diagram makes it very clear but making it rigorous in a correct manner confuses me still. –  Slugger Feb 11 '13 at 17:00
    
The only question in my mind is whether you are allowed to say that $x\in U\V$ if and only if $x\in U$ and $x\not\in V$. Perhaps you are expected to translate $U\setminus V$ first to $U\cap V^c$. In that case, the version of the answer by copper.hat is the right one to use. –  André Nicolas Feb 11 '13 at 17:06

An alternative is to do the following direct calculation:$$ \begin{align*} & (X \setminus C) \cap D = D \setminus C \\ \equiv & \;\;\;\;\;\text{"extensionality"} \\ & \langle \forall x :: x \in (X \setminus C) \cap D \;\equiv\; x \in D \setminus C \rangle \\ \equiv & \;\;\;\;\;\text{"definition of $\cap$; definition of $\setminus$, twice"} \\ & \langle \forall x :: x \in X \land x \notin C \land x \in D \;\equiv\; x \in D \land x \notin C \rangle \\ \equiv & \;\;\;\;\;\text{"logic"} \\ & \langle \forall x :: x \notin C \land x \in D \;\Rightarrow\; x \in X \rangle \\ \equiv & \;\;\;\;\;\text{"$x \in D \Rightarrow x \in X$, since $D \subseteq X$"} \\ & \mathrm{true} \\ \end{align*} $$

This shows that the assumption that $C \subseteq X$ is not necessary for this proof.

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