elementary set theory - Prove question $(Asetminus B) cup (Bsetminus C) = Asetminus C$ , $ (Asetminus B)setminus C= Asetminus(Bcup C)$

I want to prove the following statements but for do it I need some hint.

\begin{align}
\tag{1} (A\setminus B) \cup (B\setminus C) &= A\setminus C\\
\tag{2} (A\setminus B)\setminus C&= A\setminus(B\cup C)

\end{align}
Thanks!

Answer

For the first one, suppose that $(A \setminus B) \cup (B \setminus C)$ is not empty. Take any $x \in (A \setminus B) \cup (B \setminus C)$. Then either $x \in A \setminus B$ or $x \in B \setminus C$. Note that in this particular case, both cannot be true (why?). If $x \in A \setminus B$, then $x \in A$ and $x \not \in B$. If $x \in B \setminus C$, then $x \in B$ and $x \not \in C$. This does not imply that $x \in A \setminus C$. If $x \in A \setminus B$, one of the possibilities above, then this does not give us any information about whether $x \in C$.

For example, suppose $A = \{1,2,3\},\ B = \{1,2\}$, and $C = \{3\}$. Then $3 \in A \setminus B$ and so $3 \in (A \setminus B) \cup (B \setminus C)$, but $A \setminus C = \{1,2\}$ and so $3 \not \in A \setminus C$.