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Thread: Sets

  1. #1
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    Unhappy Sets

    Is there any direct formula to find the complement of (A-B)?
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  2. #2
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    Quote Originally Posted by matsci0000 View Post
    Is there any direct formula to find the complement of (A-B)?
    (complement of A) union (A intersect B)
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  3. #3
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    Complement of (A - B)

    Hello matsci0000
    Quote Originally Posted by matsci0000 View Post
    Is there any direct formula to find the complement of (A-B)?
    A slightly simpler version of the formula that HallsofIvy has given you is $\displaystyle A' \cup B$, where $\displaystyle A'$ is the complement of $\displaystyle A$ (which is sometimes written $\displaystyle A^c$).

    If you know the Laws of Sets, you can prove this as follows:

    First, the definition: $\displaystyle A - B = A \cap B'$ (the elements that are in $\displaystyle A$ but not in $\displaystyle B$)

    So $\displaystyle (A-B)' = (A \cap B')'$

    $\displaystyle = A' \cup (B')'$ (De Morgan's Law)

    $\displaystyle = A' \cup B$ (Involution Law)

    Grandad
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  4. #4
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    Thanks, Grandad, I didn't take the time to simplify.
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