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Can't solve $\left(A\setminus B\right)\setminus C=\left(A\setminus C\right)\setminus \left(B\setminus C\right)=A\setminus \left(B\cup C\right)$ any help?
I have discrete math exam and i can't quite figure out one example.
$\left(A\setminus B\right)\setminus C=\left(A\setminus C\right)\setminus \left(B\setminus C\right)=A\setminus \left(B\cup C\right)$
i tried solving it like this and i got stuck: $\left(A\setminus B\right)\setminus C=\left(A\cap \overline{B}\right)\cap \overline{C}$, what next ?
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Step 1
More precisely, we have to apply the following properties of set operations ( is A's complement):
1. De Morgan's first law: .
2. Interpretation of set difference:
Step 2
Now, we can rewrite you original equation as follows: $\left(A\setminus B\right)\setminus C=\left[\text{by prop. (2)}\right]=\left(A\cap \overline{B}\right)\setminus C=\left(A\cap \overline{B}\right)\cap \overline{C}=\phantom{\rule{0ex}{0ex}}=\left[\text{since set intersection is associative}\right]=\phantom{\rule{0ex}{0ex}}=A\cap \left(\overline{B}\cap \overline{C}\right)=\left[\text{by prop. (1)}\right]=A\cap \overline{\left(B\cup C\right)}=\left[\text{by prop. (2)}\right]=A\setminus \left(B\cup C\right).$