Mathematical analyses of digital circuits are usually performed using Boolean algebra. With it, we can simplify logic functions and minimize the number of logic gates needed in a logic network. Boolean algebra uses the addition and multiplication operation.
Subtraction and division are not permitted. Since Boolean algebra is a system of mathematics, there are fundamental laws, which can be used to evaluate them.
These laws are:
Laws of Complementation: These are the laws of inversion; it involves the change of ones to zeros and zeros to ones.
i. O = 1 and 1 = o
ii. If A = O then A = 1
iii. If A = 1 then A = 0
iv. A = A
Laws of AND Operation
v. A. O = O
vi. A.I = A
vii. A.A = A
viii. A.A = O
Laws of OR Operation
ix A + 0 = A
x. A + 1 = 1
xi A + A = A
xii. A + A = 1
Commutative laws
xiii. A + B + A
xiv. A.B = B.A
Associative laws
xv. A + (B + C) = (A + B) +C
xvi. A. (B.C) = (A.B).C
Distributive laws
xvii. A.(B+C) = (A.B) + (A.C)
xviii. A + (B.C) = (A + B) . (A + C)
Absorptive laws
xix. A+(A.B)A+B
xx. A.B + (B.C) + (A.C) = (A.B) + (B.C)
De Morgans laws
xxi. (X+Y) = X. X
xxii. (X.Y) = X + Y