Boolean Algebra

In order to deal mathematically with the digital circuits, We need to use the Boolean Algebra. It is a tool for the analysis and design of digital systems. Boolean Algebra differs in a major way from ordinary algebra because Boolean constant and variables are allowed to have only two possible values 0 or 1.

Laws of Boolean Algebra:- The three basic lows of Boolean Algebra are the same as in ordinary Algebra;

  1. Commutative Laws
  2. Associative Laws
  3. Distributive Laws

1.Commutative Laws   This Law state that

A + B = B + A

addition

AD

Law2: For multiplication:-   This law state that

A . B = B . A

and_gate_new

OR

Associative laws:-  

Law1: For Addition:-     This Law state that

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

EX

Law2: For Multiplication:-  This law state that

A ( B C ) = ( A B )C

EX

Distributive Law:- this law state that

A (B + C) = A.B + A.C

EX

Properties of Boolean Algebra:-

1. A + 0 = A2. A + 1 = 13. A . 0 = 04. A . 1 = A
5. A + A = A6. A + Ã = 17. A . A = A
8. A . Ã = 09.
10. A + AB = A
A(1+B) = A

    A(1) = A

      A  = A11. A + ÃB = A + B

        According to rule 10 put ( A+AB) in place of A

          So. A+AB + ÃB = A+B

            A+B ( A+Ã) = A+B

              A +B(1) = A+B

                A+B = A+B12. (A+B)(A+C) = A+BC

                  AA + AC + AB + BC = A+BC

                    A+AC+AB+BC = A+BC

                      A(1+C)+AB+BC = A+BC

                        A+AB+BC = A+BC

                          A (1+B) +BC = A+BC

                            A +BC = A+BCDuality Theorem:-  According to the duality Theorem a Boolean relation can be written to form another Boolean relation by

                            1.  Changing each OR sign to an AND sign.
                            2. Changing each AND sign to an OR sign
                            3. Complementing any 0 or 1 appearing in the expression.

                            For example: Distributive Law state that.

                                  A(B+C) = AB+BC

                                 A+(B.C) = (A+B).(A+C) 

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