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Math Help - need to check one of De Morgan's proofs

  1. #1
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    need to check one of De Morgan's proofs

    Hi

    I have done the proof, just need to check it. I wanted to prove that

    A\cap B=A\setminus(A\setminus B)

    Proof:

    \mbox{let}\ x\in A\cap B\ \mbox{be arbitrary}
    \Rightarrow x \in A\ \mbox{and}\ x \in B
    \Rightarrow x \in A\ \mbox{and}\ x \notin B^{c}
    \Rightarrow x \in A\ \mbox{and}\ x \notin A\setminus B
    \Rightarrow x \in A\setminus (A\setminus B)

    I know that I have to do reverse proof too. But is it ok so far ?

    Thanks
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  2. #2
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    Yes. That is correct so far.

    May I suggest that A\setminus (A\setminus B)=A\cap(A\cap B^c)^c.
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  3. #3
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    are "complements" easier to manipulate that the "setminus" notation in proofs ?
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  4. #4
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    Quote Originally Posted by issacnewton View Post
    are "complements" easier to manipulate that the "setminus" notation in proofs ?
    I think so. But each to his own.
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