# proof involving differences of sets

• October 13th 2009, 04:12 PM
proof involving differences of sets
(A-B)-C is a subset of (A-C)-(B-C)

• October 13th 2009, 04:36 PM
artvandalay11
Quote:

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