boolean algebra simplification help~

Discussion in 'Homework Help' started by minakrn, Feb 4, 2007.

  1. minakrn

    Thread Starter New Member

    Feb 4, 2007
    3
    0
    hi guys ^^;

    well.. i am new here and would greatly appreciate some help with the following algebra simplification problems (i have included my work so that i may compare the logic with you guys)
    :

    1) AC + ABC + BC

    A + C + ABC + BC -> De'Morgans
    A + ABC + BC + C -> Commutative
    A(1 + BC) + BC + C -> Distributive
    ..stuck~

    2) (A+B)(A + B)

    (A * B)(A + B) -> De'Morgans
    (A * A * B) + (A * B * B) -> Distributive
    [simplified answer]? A + B -> Identity

    3) ABC + AC

    C(A * B + A) -> Distributive
    C((A + A)(A + B)) -> Distributive
    [simplified answer]? C * (A + B) -> Identity

    4) BC + B(AD + CD)

    BC + BAD + BCD -> Distributive
    B(C + AD + CD) -> Distributive
    ..stuck~

    5) (B + C + BC)(BC + AB + AC)

    I basically tried to distribute everything.
    Cancelled using identities and got stuck~

    6) XYZ + XYZ + XYZ + XYZ

    XZ + XYZ + XYZ -> Adjacency
    XZ + X(YZ + YZ) -> Distributive
    ..stuck~ [Question: Can you simplify (YZ + YZ) to 1? (using the identity much similiar to X + X = 1)

    7) XYZ + XYZ + XYZ + XYZ

    XYZ + XYZ + XY -> Adjacency
    Z(XY + XY) + XY -> Distributive
    ..stuck~ [Question: Can you simplify (YZ + YZ) to 1? (using the identity much similiar to X + X = 1)

    Wow..that was tedious~

    Well..there you go ^^;

    Best regards~
     
  2. Dave

    Retired Moderator

    Nov 17, 2003
    6,960
    144
    Hi and Welcome to All About Circuits!

    Can I ask for one point of clarification? When you write for example AC are you meaning [ NOT(A) AND NOT(C) ] or are you meaning [ NOT(A AND C) ].

    Thanks.

    Dave
     
  3. minakrn

    Thread Starter New Member

    Feb 4, 2007
    3
    0
    Hi Dave and thanks ^^;

    Well..what I meant was:

    [ NOT(A) AND NOT(C) ] -> so it looks more like this A C.

    BTW: can you tell me what the difference is? Is it because the De'Morgans concept?


    I apologize for this confusion.~
     
  4. Dave

    Retired Moderator

    Nov 17, 2003
    6,960
    144
    To understand the difference, think about the truth tables:

    A | NOT(A) | B | NOT(B) | A.B | NOT(A.B) | NOT(A) AND NOT(B)
    --------------------------------------------------------------
    0 | 1 | 0 | 1 | 0 | 1 | 1
    0 | 1 | 1 | 0 | 0 | 1 | 0
    1 | 0 | 0 | 1 | 0 | 1 | 0
    1 | 0 | 1 | 0 | 1 | 0 | 0

    Hence NOT(A.B) is a NAND (NOT-AND) gate, whereas (NOT(A) AND NOT(B)) is a NOR (NOT-OR) gate.

    Clearer? EDIT: Apologies, the truth table hasn't come out as I expected, I recommend you write it down from the above layout but properly format the columns!

    I will have a look through your initial workings when I get a moment later.

    Dave
     
  5. minakrn

    Thread Starter New Member

    Feb 4, 2007
    3
    0
    Oh, is that what you meant?

    Well, in that case, I see where you are coming from.
     
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