Hey xods.

For your function, one decomposition is

f(A,B,C,D)=¬A¬B¬¬C+D+A¬C+¬B+¬D

= (¬A*¬B*C)+D+(A¬C)+(¬B+¬D)

= [(¬A*{¬B*C})+(A¬C)]+[(¬B+¬D)+D]

Where the () brackets are applied first and then the [] and the final result is the + of these two inputs. With regards to the the first operation involving A, B, and C, this is just two and gates with one result fed into the other (first operation is with {}).

The only thing though that needs to be asked are the following:

1) Do you need to use specific kinds of gates (usually one specific gate where you build the others and the results from these)?

2) Do you need to optimize either the expression or the circuits to use the minimum number of gates (given a certain set of gate choices)?

If the above don't apply then you can come up with a few different ways of doing the same thing that are all the right answer.

Your relationship is [(D¬A¬B¬C)+(¬A+¬D)][C+¬B] = (D¬A¬B)+(CA+C¬D)+(¬BA+¬B¬D) which doesn't quite look right.