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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