There are a few boolean identities needed to be remembered or to be understood in order to simplify boolean expression. There is a website for the boolean logic identities:
In summary, the identities are very easy to understand:
- A+A=A, if A=1 then 1+1 is 1, + is OR logic. Similarly if A=0, 0+0=0.
- A*A=A, same as (1), just to confuse someone.
- A+1=1, anything OR 1 is 1, cannot be confused.
- A+0=A, 0 is something you can forget.
- A*1=A, * is boolean multiplication, AND operation. 1 AND anything depends on anything.
- A*0=0, 0 will make anything 0, like regular multiplication.
- (A’)’=A, ‘ is called prime, meaning NOT, NOT NOT will cancel out, double negative.
- A+A’=1, obviously as 0+1 or 1+0 will give you 1
- A*A’=0, same, 0*1 or 1*0 will be 0.
- A+A*B=A, or A+AB, is not too easy to see, but factor A out you will have 1+B, ie A(1+B)=A*1=A
- A(A+B)=A, or A*(A+B), similarly, you can multiply it to be AA+AB=A+AB, same as before equals A.
- A+A’B=A+B, if A=0, A+A’B=B, if A=1, A+A’B=1=A, so obviously. This is called Nashelsky’s theorem
- A(A’+B)=AB is easy to see. Expanding to have AA’+AB=0+AB=AB. This is also called Nashelsky’s theorem.
- (A+B)(A+C)=A+BC, expanding to have AA+AC+AB+BC = A+AC+AB+BC = A+AB+BC = A+BC using property (10).
- (AB)’=A’+B’ for DeMorgan’s theorem
- (A+B)’=A’B’ for the other DeMorgan’s theorem.