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Thread: proof involving differences of sets

  1. #1
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    proof involving differences of sets

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

    please help me prove this please
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  2. #2
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    Quote Originally Posted by leinadwerdna View Post
    (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, $\displaystyle (A-B)-C\subseteq (A-C)-(B-C)$
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