# Math Help - proof involving differences of sets

1. ## proof involving differences of sets

(A-B)-C is a subset of (A-C)-(B-C)

please help me prove this please

2. Originally Posted by leinadwerdna
(A-B)-C is a subset of (A-C)-(B-C)

please help me prove this please
Let $x\in (A-B)-C$. So $x\in (A-B)$ and $x\not\in C$ by applying definition

So $x\in A$ and $x\not\in B$ and $x\not\in C$ by applying definition

Therefore, $x\in (A-C)$ and $x\not\in (B-C)$, so by applying the definition in reverse, this means $x\in (A-C)-(B-C)$

So we showed that if we have an element of $(A-B)-C$, then it is also in $(A-C)-(B-C)$

In other words, $(A-B)-C\subseteq (A-C)-(B-C)$