exercise sheet 1 question 3.b: Equations to Clause Form for Basic Operators

Question

Convert the following equations into conjunctive normal form.

I tried to solve this question in 2 ways.

once with "Multiplying-out":

and also with "Shannon decomposition":

but each time I got a different answer and I don't know why. Would you please tell me what I did wrong in  Shannon decomposition?

Answer 1

The correct rule with Shannon decomposition is 

    (x | phi[x<-0]) & (!x | phi[x<-1]) 

and using it you get

     (x <-> (y1 | y2)) 
     <-> ((x | !(y1 | y2)) & (!x | (y1 | y2)) )
     <-> (x | !y1 & !y2) & (!x | y1 | y2) 
     <-> (x | !y1) & (x | !y2) & (!x | y1 | y2)