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

Question

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

Answer 1

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