# Minimization

BOOLEAN ALGEBRA LAWS

(i) A + A = A

(ii) A . A = A

(i) A + B = B + A

(ii) A . B = B . A

(i) A + ( B + C) = ( A + B ) + C = A + B + C

(ii) A . (B . C ) = ( A . B) C = A B C

(i) A ( B + C ) = AB + AC

(ii) A + ( BC ) = ( A + B ) ( A + C )

(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

(i) A . A = A

(ii) A .0 = 0

(iii) A . 1 = A

(iv) A . Ä€ = 0

( A' )' = A'' = A

(i) A + A = A

(i) A + 0 = A

(iii) A + 1 = 1

(iv) A + Ä€ = 1

(i) ( A1 . A2 . A3 . ...... An)'' = Ä€1+ Ä€2 + Ä€3 + ........ + Ä€n

(ii) ( A1 + A2 + A3 + ....... + An)'' = Ä€1 . Ä€2 . Ä€3 .....Ä€n

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

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

"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

**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 . Ä€ = 0

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

**OR - operation Theorem**(i) A + A = A

(i) A + 0 = A

(iii) A + 1 = 1

(iv) A + Ä€ = 1

**De Morgan's Theorem**(i) ( A1 . A2 . A3 . ...... An)'' = Ä€1

(ii) ( A1 + A2 + A3 + ....... + An)'' = Ä€1 . Ä€2 . Ä€3 .....Ä€n

**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

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