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Math Help - Boolean Algebras

  1. #1
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    Boolean Algebras

    Write the dual of each statement.
    (x + y)(x+1) = x + xy + y

    As the books skips all of the in between steps, i can only postulate on what I'm supposed to do. (I'm thinking of using a supplemental book because this book isn't helpful)
    Since the book also uses a example as a definition, I'm guesing that the dual of an equation is one that is fundamentally the same but in a different form.

    What I did:
    (x+y)(x+1) = x + xy + y
    x^2 + x + xy + y = x(y+1) + y

    However, the answer is  xy + x0 = x(x+y)y

    The professor lost me in the lecture, though I hope to meet with him later to work on some other problems.

    Thanks!
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  2. #2
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    Yes, when a book uses an example as a definition, it's frustrating. I think that "dual" means exchanging "and" with "or" and 0 with 1, as well as negating every variable. So the dual of (x+y)(x + 1) is \bar{x}\bar{y}+\bar{x}0. I also think that the original equation has implicit universal quantifiers:

    \forall x\forall y\,(x+y)(x+1) = x + xy + y

    i.e., it is viewed as a univesal law, as opposed to an equation that one has to solve. When universally quantified,

    \bar{x}\bar{y} + \bar{x}0 = \bar{x}(\bar{x}+\bar{y})\bar{y}

    is equivalent to

    xy + x0 = x(x+y)y

    Of course, negating variables may not be a part of this particular definition of "dual".
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