# How to convert SoP to NOR?

• I have a logical expression: F = AB + AC + CB
• I know its equivalent expression via NOR: R = (A ⊽ B) ⊽ (A ⊽ C) ⊽ (C ⊽ B)
• It is assumed that the desired expression in NOR is unknown.
• I know De Morgan's laws, and that the element NOR is equivalent to the expression !P!Q!S, which is equivalent to the expression !(P + Q + S). But I can't get PoS expression through double negation. For example: !( (!A+!B)(!A+!C)(!C+!B) )

How to prove that in order to convert SoP to NOR, it is desirable to convert SoP to PoS first? How to bring the expression F to R?

• How does one prove desirable? ("How to prove that in order to convert SoP to NOR, it is desirable to convert SoP to PoS first?") In any case, It just sounds to me as though you need some sit-down time for playing. You need to expand everything and rearrange. See this answer I gave to a different question. It's related, though you may not yet see why. You couldn't use double negation because you didn't include all the terms and didn't then rearrange stuff from there. If you still need help I'll write something if no one else bothers.
– jonk
Nov 19, 2022 at 5:57
• Meanwhile, I'll +1 the question to see if it gets the attention of my betters. (Despite the fact that I don't like to see desirable as the object of proof in the same sentence.) One more thing to remember, imagination is vital. And that means NOT mechanical 'follow the rules' codified, straight forward application of all the stuff you are being taught. It means being able to climb out of the well-worn paths you are trained to use. To move beyond them, climb out into unknown and unexplored ways of thinking. Work on that, too. Use your imagination. Develop it.
– jonk
Nov 19, 2022 at 6:03
• If it's not clear already you need to see that A B = A B C + A B !C, that A C = A B C + A !B C, and that B C = A B C + !A B C. Use those to replace your original equation. Re-arrange the results and move towards the solution you desire. Imagine!! This is the time to stretch yourself.
– jonk
Nov 19, 2022 at 6:15
• Very good. Now use your imagination! The double negation gets you directly and unimaginatively towards !( (!A + !B) (!A + !C) (!B + !C) ). Which you don't want. Think about how to re-arrange things so you get where you want to be!!! It is there in front of you! Perhaps read the link I gave. It may help. But you can do this! I really do NOT want to steal this process from you. Stuff handed on a silver platter doesn't stick as well.
– jonk
Nov 19, 2022 at 6:35
• Remember that if AB= ABC+AB!C ... then it says that ABC is "included" in AB ... so you don't need ABC when you find an expression as AB+ABC = AB(1+C)= AB ... Nov 19, 2022 at 11:33