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Boolean Algebra

Duality Principle:

This property of Boolean algebra state that all binary expressions remain valid when following two steps are performed:

Step 1: Interchange OR and AND operators.

Step 2: Replace all 1’s by 0’s and 0’s by 1’s.

Huntington Postulates

Adding a number to ‘0’ to itself

P1. Postulate:- x + 0 = x

From duality of P2

P2. Postulate:- x * 1 = x

Sum of a number and its complement from a Set is ‘1’ 

P3. Postulate:- x + x’= 1

From duality of P3

P4. Postulate:- x * x’ = 0

From commutative property of binary numbers we have

P5. Postulate:- x + y = y + x

From duality of P5

P6. Postulate:- x * y = y * x

From distributive property of binary numbers we have

P7. Postulate:- x(y + z) = xy + xz

From duality of P7

P8. Postulate:- x + yz = (x + y) (x + z)

Theorems of Boolean Algebra derived from Huntington postulates - Discussion

T1. Theorem:- x + x = x

x + x = (x + x)*1= (x + x)(x + x’)

From P8, x + xx’ = x

From duality of T1

T2. Theorem:- x*x = x

T3. Theorem:- x + 1 = 1

x + 1= (x + 1).1= (x +1)*(x + x’)

From P8, (x + 1*x’)= (x + x’)= 1

From duality of T3

T4. Theorem:- x.0 = 0

T5. x + (y + z) = (x + y) + z

From duality of T5

T6. Theorem:- x(yz) = (xy)z

T7. Theorem:- (x’)’ = x

From duality of T10

T11. Theorem:- x(x+y) = x

T8. Theorem:- (x + y)’ = x’y’

T9. Theorem:- (xy)’ = x’ + y’

T10. Theorem:- x + xy = x