I am working on a textbook exercise that asks to "Minimize the following boolean expression using boolean algebraic laws, and then develop a logic circuit for it WITHOUT using a NOT gate".
The expression is X= (NOT A) AND (NOT B OR C) OR NOT(C OR A)
First, I attempt to minimize this equation. I use distribute law of the left hand side to turn (NOT A) AND (NOT B OR C) to (NOT A AND NOT B) OR (NOT A AND C).
So following this law, the new equation is (NOT A AND NOT B) OR (NOT A AND C) OR (NOT C AND NOT A).
Grouping at the right hand side of ((NOT A AND C) OR (NOT C AND NOT A)), shows that there is another law that states (AB + A(NOT B) = A).
I use this LAW (Call this law ?) here to simplify the entire expression to simply NOT(A).
So the new equation is (NOT A AND NOT B) OR (NOT A).
I use the redundancy law, which states that (A + AB = A), to change the entire expression to simply NOT(A).
Following this minimization, I have successfully minimized the entire equation to X=NOT(A). Truth tables show they are functionally equivalent.
My two questions are 1. The area where I saw ? Law, I have searched for a name for this law but have not been able to find one. Is there a name for the law (AB + A(NOT B) = A)? 2. If the function is NOT(A), I am not sure how one could design a logic circuit without using a NOT gate? I was thinking that maybe a NAND gate could work, but those have inverters. If it asks to not use a NOT gate, is it still valid to use a NAND gate?
Thank you for any help.
There is no name for AB + A(NOT B) = A, the equation is just a consequence of the distribution law AB + A(NOT B) = A(B + NOT B) and the fact that B + NOT B = TRUE.
Regarding your concern about NAND, yes you need to used it because there is no way to get NOT from just OR and AND.