Boolean Identities for NOT Operation:
The NOT operation of a Boolean variable A is indicated by its complement A. Following are the Boolean identities pertaining to NOT operation.
A¯¯ = A
This implies that the double complement of a Boolean variable is the variable itself. For A = 0, A = 1, A¯¯ = 0 and for A = 1, A = 0, A¯¯ = 1 is true.
A + A = 1
This means that a variable ORed with its complement always equals 1. If A = 0, A = 1 and A + A = 1 and when A = 1, A = 0 and A + A = 1 is exact.
A A = 0
This means that a variable ANDed with its complement always equals 0. For two possible values of A, 0.1 = 0 while A = 0, 1.0 = 0 whereas A = 1 is true.
A + A B = A + B
Proof: A + A B = A (B + 1) + A B = A B + A + A B = (A + A) B + A = B + A
= A + B