digital-logickarnaugh-mapnor

How can w’xz + w’yz + x’yz’ + wxy’z be implemented with 4 NOR gates (+ inverters), given d = wyz


How can you implement the following function with only four NOR gates and inverters:

F = w’xz + w’yz + x’yz’ + wxy’z

d = wyz


Solution

  • First get the Karnaugh map:

    w’x z + w’ yz +  x’yz’ + wxy’z
    a       b         c        e
    
    wx\yz 00  01  11  10
    
                ----   c
    00    0   0 | 1| |1 |
                | b| ----
            ----|----
    01    0 |a1 | 1|| 0
            ----|--|-
                ---
    11    0   1e  0   0 
                     
                     ---
    10    0   0   0  |1 |
    

    Then get the Product Of Sums:

    wx\yz 00  01  11  10
    
        |      |
        |      |
    00  | 0   0|   1   1 
        --------
                
      ------         -------
    01    0|  1    1 | 0
           |         |   A
           |      ---|
    11    0|  1   |0|| 0 
      ------      |C|--------
       --------   | |  
    10 |  0   0|  |0|  1
       |       |  ---
       |       |
       |  B    |
     
    (x'+z)(x+y)(w'+y'+z')
      A      B       C
    

    Then reduce to utilize the available d =wyz pin, and to get the requested 4 nors:

    =( (x'+z)(x+y) (w'+y'+z'))''
    =(((x'+z)(x+y))(w'+y'+z'))''
    =(((x'+z)(x+y))'+(w'+y'+z')')'
    =(((x'+z)(x+y))'+(wyz))'
    =(((x'+z)(x+y))'+d)'
    =(((x'+z)'+(x+y)')+d)'
    =(((x'+z)'+(x+y)')''+d)'
             ^      ^ ^    ^  4 nors
         ^             ^      2 inverters