Laws of Boolean Algebra

Boolean Algebra
The most obvious way to simplify Boolean expressions is to manipulate them in the same way as normal algebraic expressions are manipulated. With regards to logic relations in digital forms, a set of rules for symbolic manipulation is needed in order to solve for the unknowns.
A set of rules formulated by the English mathematician George Boole describe certain propositions whose outcome would be either true or false. With regard to digital logic, these rules are used to describe circuits whose state can be either, 1 (true) or 0 (false). In order to fully understand this, the relation between the AND gate, OR gate and NOT gate operations should be appreciated. A number of rules can be derived from these relations as Table 1 demonstrates.

P1: X = 0 or X = 1
P2: 0 . 0 = 0
P3: 1 + 1 = 1
P4: 0 + 0 = 0
P5: 1 . 1 = 1
P6: 1 . 0 = 0 . 1 = 0
P7: 1 + 0 = 0 + 1 = 1
Laws of Boolean Algebra

the basic Boolean laws. Note that every law has two expressions, (a) and (b). This is known as duality. These are obtained by changing every AND(.) to OR(+), every OR(+) to AND(.) and all 1's to 0's and vice-versa.
It has become conventional to drop the . (AND symbol) i.e. A.B is written as AB.

T1 : Commutative Law

(a) A + B = B + A
(b) A B = B A

T2 : Associate Law

(a) (A + B) + C = A + (B + C)
(b) (A B) C = A (B C)

T3 : Distributive Law

(a) A (B + C) = A B + A C
(b) A + (B C) = (A + B) (A + C)

T4 : Identity Law

(a) A + A = A
(b) A A = A

T5 :

(a)
(b)

T6 : Redundance Law

(a) A + A B = A
(b) A (A + B) = A

T7 :

(a) 0 + A = A
(b) 0 A = 0

T8 :

(a) 1 + A = 1
(b) 1 A = A

T9 :

(a)
(b)

T10 :

(a)
(b)

T11 : De Morgan's Theorem

(a)
(b)

Examples

Prove T10 : (a)

(1) Algebraically:

(2) Using the truth table:

Using the laws given above, complicated expressions can be simplified.