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DeMorgans Laws formulation

  1. Aug 8, 2014 #1
    Hello,
    I am working through Munkres Topology (not for a class). It asks the reader to formulate and proove DeMorgans Laws. I am new to proofs, so I was wondering if this is what the book is asking. Any help would be appreciated!

    assume two sets
    [itex]\, \,\begin{Bmatrix}
    A-(B\cup C)\,
    \end{Bmatrix}\, and\, \begin{Bmatrix}
    (A-B)\cup (A-C)
    \end{Bmatrix}\, \, \, [/itex]

    [itex]\forall x\in (B\cup C), x\in B\, or\, x\in C \, or \, both[/itex]

    [itex]\therefore \, \, \forall x\, \in\begin{Bmatrix}
    A-(B\cup C)\,
    \end{Bmatrix}, x\in A[/itex]

    Now,

    [itex]\forall x\in (A-B), \, x\in A\, \, and\, \, \forall x\in (A-C), \, \, x\in A[/itex]

    [itex]\Rightarrow \forall x\in \begin{Bmatrix}
    (A-B)\cap (A-C), \, x \in A
    \end{Bmatrix}[/itex][itex] x \in A[/itex]


    Thus: [itex]\begin{Bmatrix}
    A-(B \cup C)
    \end{Bmatrix}
    =\begin{Bmatrix}
    (A-B)\cap (A-C)
    \end{Bmatrix}[/itex]

    Does this proove DeMorgans Law (just the first one)? Formally?

    Thanks again!
     
    Last edited: Aug 8, 2014
  2. jcsd
  3. Aug 8, 2014 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    If [itex]x\in[/itex]"A- anything" then [itex]x\in A[/itex]. You haven't said anything about x NOT being in the other sets.
     
  4. Aug 8, 2014 #3
    so I also need to state: [itex] x\notin B[/itex] and [itex] x\notin C[/itex] ?
    So if I tagged those statements onto the proof after I state [itex]x\in A[/itex], then its good?

    Im just a little confused, by saying:

    [itex]\forall x\in (B\cup C), x\in B\, or\, x\in C \, or \, both[/itex]

    [itex]\therefore \, \, \forall x\, \in\begin{Bmatrix}
    A-(B\cup C)\,
    \end{Bmatrix}, x\in A[/itex]

    Aren't I stating that I am taking out all elements of B and C, thus all elements left are elements of A?
     
    Last edited: Aug 8, 2014
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