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