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Homework Help: Deriving full adder sum and carry outputs using boolean algebra

  1. Apr 16, 2009 #1
    1. The problem statement, all variables and given/known data

    Hi, I am trying to write the sum and output of a full adder in terms of XOR logical functions using boolean logic and Karnaugh maps. I've got the expressions from the Karnaugh maps fine but I can't seem to rearrange them into the expected form shown at the end of my working.

    2. Relevant equations

    Explained above.

    3. The attempt at a solution

    It's going to be difficult writing my working here but hopfully it is clear:

    My equation obtained from the Karnaugh map using a minimized SOP:
    Sum = NOT(A).NOT(B).C + NOT(A).B.NOT(C) + A.B.C + A.NOT(B).NOT(C)
    Sum = NOT(A).(NOT(B).C + B.NOT(C)) + A.(B.C + NOT(B).NOT(C))
    Sum = NOT(A).(B XOR C) + A.(B.C + NOT(B).NOT(C))


    Sum = NOT(NOT(A).B + A.NOT(B)).C + (NOT(A).B + A.NOT(B)).NOT(C)
    Sum = (A XOR B) XOR C

    This is where I'm trying to get. I've tried going both backwards and forwards but I just don't know where to go in-between. Any suggests to get me past the 3rd step I'm at?

    Thank you,
  2. jcsd
  3. Apr 17, 2009 #2
    I've made some significant progress but I've become stuck again.. here is what I've got:


    Attached Files:

    • sum.png
      File size:
      3.2 KB
  4. Apr 17, 2009 #3


    User Avatar

    Staff: Mentor

    Ouch. Welcome to the PF. This is a 3-bit full-adder? Could you post the truth table, including the carry bit? But then how do K-maps help in an XOR implementation of the truth table? I don't think I've ever had to morph into XOR logic (must be an academic thing?) -- what are the techniques for going from traditional minterm implementations (K-maps) to XOR logic?
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