SYNTHESIS OF SWITCHING CIRCUITS 71 



(3) Now, Still assuming we have one function, VZ' -\- Y'Z, from E, 

 suppose at least two of the remaining are from D. Using a similar "tapping 

 off" process we can save an element on each of these. For instance, if the 

 functions are Y + Z and I'' + Z' the circuit would be as shown in Fig. 11. 



(4) Under the same assumption, then, our worst case is when two of the 

 functions are from C and one from D, or all three from C. This latter case 

 is satisfactory smce, then, at least one of the three must be a term of 

 YZ' -\- Y'Z and can be "tapped off." The former case is bad only when 

 the two functions from C are YZ and Y'Z'. It may be seen that the only 



» o o- 



Z' 



.b 



Y' 



Fig. 11 — Simplifying network. 



Fig. 12 — Simplifying network. 



essentially different choices for the function from D are Y -{- Z and Y' + Z. 

 That the four types of functions/ resulting may be realized with 14 elements 

 can be shown by writing out typical functions and reducing by Boolean 

 Algebra. 



(5) We now consider the cases where two of the functions are from E. 

 Using the circuit of Fig. 12, we can tap off functions or parts of functions 

 from A, B or D, and it will be seen that the only difficult cases are the fol- 

 lowing: (a) Two functions from C. In this case either the function/ is 

 symmetric in F and Z or else both of the two functions may be obtained 

 from the circuits for the E functions of Fig. 12. The symmetric case is 

 handled in a later section, (b) One is from C, the other from D. There 

 is only one unsymmetric case. We assume the four functions are Y © Z, 

 Y @ Z', YZ and Y + Z'. This gives rise to four types of functions /, 

 which can all be reduced by algebraic methods. This completes the proof. 



