Sorry, could not post the notes as scheduled, was little busy with other thing. Today giving some Boolean Laws and Theorems. Boolean logic was developed by an English mathematician George Boole, used to construct and solve Boolean algebra and helped to design the electrical circuits.
The basic laws of Boolean algebra: | |||||||||||
Proof (Using Truth Table)
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A + A.B = A
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A( A + B) = A
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A
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B
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A.B
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A+A.B
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A
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B
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A+B
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A.(A+B
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0
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0
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0
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0
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0
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0
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0
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0
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0
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1
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0
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0
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0
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1
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1
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0
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1
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0
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0
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1
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1
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0
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1
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1
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1
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1
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1
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1
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1
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1
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1
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1
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Proof (Using Postulate)
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If A and B are elements of a Boolean algebra
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A+A.B = A
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A( A + B) = A
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A.1+A.B
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A.1 = A
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A.A + A.B
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Distributive Law
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A(1 + B)
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Distributive Law
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A + A.B
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Idempotent law
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A.1
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Law on the property
of 1 |
A.1 + A.B
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A.1 = A
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A
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A.1 = A
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A(1 + B)
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A.1
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A
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A
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Now just tell you about principle of duality.
Principle of Duality
The dual of any statement in a Boolean algebra is the statement obtained by interchanging OR with AND, and simultaneously inter-changing the elements 0 and 1 in the statement.
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IdentityBoolean
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Dual
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A + 0 = A
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A.1 = A
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A + 1 = 1
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A.0 = 0
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A + A = A
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A.A=A
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A + A’ = 1
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A.A’ = 0
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A + B = B + A
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A . B = B . A
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A+ B.C = (A + B)(A + C)
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A(B + C) = A . B + A. C
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A( A + B) = A
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A+A.B=A
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A + AB = A + B
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A.(A+B) = A
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(A+ B)’ = A’.B’
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(AB)’=A’ + B’
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DeMorgan’s theorems provide mathematical verification of: :
1. The equivalency of the NAND and negative-OR gates 2. The equivalency of the NOR and negative-AND gates. | ||||||||||||||
(A.B)' =
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A'+B'
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(A + B)' =
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A'.B'
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**Change the sign and break the line.
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(A.B)' = A'+B'
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(A + B)' = A'.B'
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A
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B
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A.B
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(A.B)'
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A'
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B'
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A'+B'
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A
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B
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A+B
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(A+B)'
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A'
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B'
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A'.B'
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0
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0
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0
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1
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1
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1
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1
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0
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0
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0
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1
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1
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1
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1
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0
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1
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0
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1
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1
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0
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1
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0
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1
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1
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0
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1
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0
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0
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1
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0
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0
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1
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0
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1
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1
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1
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0
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1
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0
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0
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1
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0
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1
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1
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1
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0
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0
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0
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0
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1
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1
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1
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0
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0
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0
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0
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Next time it will be time for Logic circuits using Logic Gates.
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!!!Wish all of you a Shuvo Vijaya Dashami / Happy Dussera!!!
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