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

please help me prove this please

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- Oct 13th 2009, 03:12 PMleinadwerdnaproof 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 PMartvandalay11
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, $\displaystyle (A-B)-C\subseteq (A-C)-(B-C)$