(A-B)-C is a subset of (A-C)-(B-C)
please help me prove this please
Let $\displaystyle x\in (A-B)-C$. So $\displaystyle x\in (A-B)$ and $\displaystyle x\not\in C$ by applying definition
So $\displaystyle x\in A$ and $\displaystyle x\not\in B$ and $\displaystyle x\not\in C$ by applying definition
Therefore, $\displaystyle x\in (A-C)$ and $\displaystyle x\not\in (B-C)$, so by applying the definition in reverse, this means $\displaystyle x\in (A-C)-(B-C)$
So we showed that if we have an element of $\displaystyle (A-B)-C$, then it is also in $\displaystyle (A-C)-(B-C)$
In other words, $\displaystyle (A-B)-C\subseteq (A-C)-(B-C)$