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 viceversa.
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
Top
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.
