Minimization

BOOLEAN ALGEBRA LAWS

Identity Law
     (i)  A + A = A
     (ii) A . A = A
   
Commutative Law
     (i) A + B = B + A
     (ii) A . B = B . A

Associative Law
    (i) A + ( B + C) = ( A + B ) + C  = A + B + C
    (ii) A . (B . C ) = ( A . B) C  = A B C

Distributive Law
   (i) A ( B + C ) = AB + AC
   (ii) A + ( BC ) = ( A + B ) ( A + C )

De Morgan's Law
   (i) ( A + B )' = A' B'
   (ii) ( AB)' = A' + B'

NAND and NOR gates do not follow associative law but follow commutative law.
OR , AND , EXOR , EXNOR follow commutative and associative both laws.
A + AB = A
A ( A + B ) = A
A + A'B = A + B
A ( A' + B ) = AB


 

BOOLEAN ALGEBRAIC THEOREMS

AND-operation Theorem
   (i) A . A = A
   (ii) A .0 = 0
   (iii) A . 1 = A
   (iv) A . Ā = 0

Involution Theorem
    ( A' )' = A'' = A

OR - operation Theorem
    (i) A + A = A
    (i) A + 0 = A
    (iii) A + 1 = 1
    (iv) A + Ā = 1

De Morgan's Theorem
   (i) ( A1 . A2 . A3 . ...... An)'' =  Ā1 + Ā2 + Ā3 + ........ + Ān 
   (ii) ( A1 + A2 + A3 + ....... + An)'' = Ā1 . Ā2 . Ā3 .....Ān 

Transposition Theorem
   ( A + B ) ( A + C ) = A + BC

Distribution Theorem
   A + BC = ( A + B ) ( A + C ) 

Boolean Algebraic Theorems

Duality Theorem
 "Dual expression" is equivalent to write to write a negative logic of the given Boolean relation.for this we:
 (i) Change each OR sign by an AND sign and vice-versa.
 (ii) Complement any '0' or '1' appearing in expression.
 (iii) Keep literals as it is.
For 1-time Dual , it is called "Self Dual Expression".
For n-variable maximum possible Self - Dual Function