Given that $A$, $B$ and $C$ are sets, and that $(A ∪ B) \setminus C ⊆ A \setminus B$.
Prove that $(A \setminus C) ∩ B = ∅$.
I tried to prove it this way:
It is given that the containing set $(A \setminus B)$ doesn't contain any $x \in B$ (by the definition of set difference). Therefore, no $x \in B$ & $x \notin C$ exists. Therefore, in no way there is $x \in B$ & $x \notin C$ & $x \in A$ exists (equivalent to $(A \setminus C) ∩ B = ∅ )$.
I'm afraid my proof is incorrect, because $x \notin A$ seems unneccesary.
I'm a new student in the university in Israel, learning parallel to my highschool studies. English isn't my mother tongue. I am sorry for any mistakes and my bad formatting.
Thank you!
http://www.freeimagehosting.net/newuploads/i378c.png
(Added by A.K. translation of the Hebrew in the image)
By the assumption every $x$ which belongs to $A$ or $B$, and does not belong to $C$ is necessarily an element of $A$ and not an element of $B$ (by the definitions of inclusion, union and difference). In the right hand side ($A\setminus B$) no element belongs to $B$, therefore it is impossible that in the left hand side there is an element which belongs to $B$. Therefore there is no $x\in B$ and $x\notin C$. In particular there is no such $x$ for which $$x\in A\land x\in B\land x\notin C$$ Q.E.D
Best Answer
You have to prove that $A\setminus C$ is disjoint from $B$ (i.e. their intersection is empty), when you are given that $(A\cup B)\setminus C\subseteq A\setminus B$.
We need to show that if $x\in A\setminus C$ then $x\notin B$, and if $x\in B$ then $x\notin A\setminus C$. If we have shown both these things then there is no $x$ which is an element of both $A\setminus C$ and $B$, therefore $(A\setminus C)\cap B=\varnothing$.
Given $x\in B$ we know that $x\notin A\setminus B$ by the definition of $\setminus$. In particular this means that $x\notin (A\cup B)\setminus C$ (because the assumption was that $(A\cup B)\setminus C\subseteq A\setminus B$).
Therefore either $x\notin A\cup B$ or $x\in C$. Since we assume $x\in B$ we have to have that $x\in C$ as well. Therefore $x\notin A\setminus C$.
On the other hand, if $x\in A\setminus C$ we have that $x\in A$ and $x\notin C$. In particular this means that $x\in A\cup B$ and $x\notin C$ and therefore $x\in(A\cup B)\setminus C$. The assumption tells us, again, that $x\in A\setminus B$. Therefore $x\notin B$.