# proof involving differences of sets

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• Oct 13th 2009, 03:12 PM
leinadwerdna
proof involving differences of sets
(A-B)-C is a subset of (A-C)-(B-C)

please help me prove this please
• Oct 13th 2009, 03:36 PM
artvandalay11
Quote:

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, $(A-B)-C\subseteq (A-C)-(B-C)$