# Laws of Boolean algebra

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

not allowed!