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Concept of Dual!?

Discussion in 'Electronic Basics' started by Steven O., Oct 17, 2004.

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  1. Steven O.

    Steven O. Guest

    I'm just learning digital electronics, taking an introductory class.
    Everything is pretty clear so far, except the concept of a "dual" of a
    logical function has me slightly puzzled.

    I think it's like this: If I take any true logical equation, and
    reverse the operators, and interchange 1s and 0s, (and, do NOT switch
    A to A' or vice-versa), the equation I get as a result is still true,
    and is the dual of the original -- but the new equation is NOT the
    equivalent of the orginal. Is that right?

    For example:

    A + A' = 1
    AA' = 0

    Or another example:

    A + 1 = 1
    A0 = 0

    Or, once more:

    A + 0 = A
    A1 = A

    Are each of these pairs, in fact, the dual of each other? Thanks in
    advance for all replies.

    Steve O.

    "Spying On The College Of Your Choice" -- How to pick the college that is the Best Match for a high school student's needs.
  2. Ratch

    Ratch Guest

    Duality: Every Boolean expression remains valid if the operations and
    identity elements are interchanged.

    Ex: If (x+y)' = x' * y' holds, then (x * y)' = x' + y' also holds.

    Ex: If x + 1 = 1 holds, then x * 0 = 0


    the Best Match for a high school student's needs.
  3. To prove AA' = 0 from A + A' = 1 use DeMorgan
    (A + A')' = 1'
    A' A" = 0
    A' A = 0
    AA' = 0
    (A + 1)' = 1'
    A' 1' = 0
    A' 0 = 0

    but as this is for all A, we have for all A
    (A')' 0 = 0
    A0 = 0
    (A + 0)' = A'
    A' 0' = A'
    A' 1 = A'

    again as this is for all A,
    A" 1 = A"
    A1 = A

    Yes, and now you know the potency of DeMorgan's rules and how
    to derive the dual of any universal equation.
  4. Michael Barr

    Michael Barr Guest

    These are NOT (with one exception) the dual equations, although they
    are all valid. The dual expression is gotten by also complementing
    each variable. So the dual of A + 0 = A is A'1 = A'. The dual of A +
    B = C is A'B' = C'. Note that the latter equation is not a tautology,
    which all of your examples were. Now it is a characteristic of a
    tautology that if you replace each free variable by an arbitrary
    expression, you still get a tautology. So if you begin with A + 0 = A
    and replace A by A', you get the tautology A' + 0 = A'. If you now
    dualize, you get A1 = A, again a tautology. Now A + B = C is a
    contingent expression; its truth depends on those of A, B and C, but A
    + B = C for exactly the same values as A'B' = C' and NOT those that
    make AB = C true.

    So to repeat, to dualize an expression (or an equation), exchange 0
    and 1, exchange meet and join and complement each variable.
  5. Steven O.

    Steven O. Guest

    Thank you.

    Steve O.

    "Spying On The College Of Your Choice" -- How to pick the college that is the Best Match for a high school student's needs.
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