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Electronics
Electronics Catlog
DIC Catlog

Number System
Conversions Between Number System
Arithematic Operations
1's & 2's Complement
Gray Codes
Arithmetic Circuits
Logical Gates and Truth Table Funtions
Boolean Expressions
Boolean Algebra
Karnaugh Map
Multiplexer
DeMultiplexer
Encoder & Decoder
TTL Circuits
Multivibrators
555 Timer
Flip Flops
RS Flip - Flop
JK Flip - Flop
D Flip - Flop
Shift Register
Schmitt Trigger
Asynchronous Counters
Synchronous Counters
Digital - Analog Conversion
Data Flow
ROM
Memory Drives
Electronics Equation
Resistor Color Codes

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 demonstrated.


Boolean Postulates


    * 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 are below. 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.

Commutative Law


    (a) A + B = B + A

    (b) A B = B A

Associate Law


    (a) (A + B) + C = A + (B + C)

    (b) (A B) C = A (B C)

Distributive Law


    (a) A (B + C) = A B + A C

    (b) A + (B C) = (A + B) (A + C)

Identity Law


    (a) A + A = A

    (b) A A = A

T5 :

    (a) Boolean Algebra ExT5a

    (b) Boolean Algebra ExT5b

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) Boolean Algebra ExT9a

    (b) Boolean Algebra ExT9b

T10 :

    (a) Boolean Algebra ExT10a

    (b) Boolean Algebra ExT10b

De Morgan's Theorem


    (a) De Morgan's Law 01

    (b) De Morgan's Law 02