Boolean Algebra and Logic Simplification

Boolean Algebra and Logic Simplification

Boolean Algebra and Logic Simplification

 

LOGIC SIMPLIFICATION

1) LAWS AND RULES OF BOOLEAN ALGEBRA

=> Laws of Boolean Algebra

Commutative Laws

1) The commutative law of addition for two variables is written as

A+B = B+A

               Fig 1: Application of commutative law of addition.

2)The commutative law of multiplication for two variables is A.B = B.A

                    Fig.2 Application of commutative law of multiplication

 

Associative Laws:

1. The associative law of addition is written as follows for three variables:

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

               Fig.3 Application of associative law of addition.

2.The associative law of multiplication is written as follows for three variables:

                A(BC) = (AB)C

                          Fig.4 Application of associative law of multiplication.

 

 

Distributive Law:

1.The distributive law is written for three variables as follows:

A(B + C) = AB + AC

                                             Fig.5 Application of distributive law.

 

Basic rules of Boolean algebra.

 

Rule 1. A + 0 = A

 

Rule 2. A + 1 = 1

 

 

Rule 3. A . 0 = 0

 

Rule 4. A . 1 = A

 

 

Rule 5. A + A = A

 

 

Rule 6. A + A = 1

 

Rule 7. A . A = A

 

Rule 8. A . A = 0

 

Rule 9 A = A

 

 

Rule 10. A + AB = A

This rule can be proved by applying the distributive law,

 

A + AB = A( 1 + B)                        distributive law

 = A . l                                             (1 + B) = 1

 = A                                                 A . 1 = A

 

The proof is shown in Table below  which shows the truth table and the resulting logic circuit simplification.

Rule 11. A + AB = A + B

Take LHS

A + AB = (A + AB) + AB              since A = A + AB

             = (AA + AB) + AB            since A = AA

             =AA +AB +AA +AB         adding AA = 0

             = (A + A) (A + B)              Factoring = 1. (A + B)

             =A + B                               A + A = 1

The proof is shown in Table 4-3, which shows the truth table and the resulting logic circuit simplification.

 

Rule 12. (A + B)(A + C) = A + BC

Take LHS

 (A + B)(A + C) = AA + AC + AB + BC                  Distributive law

                        = A( 1 + C) + AB + BC             

                        = A. 1 + AB + BC                            1 + C = 1 Factoring (distributive law)

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

                         = A + BC                                          A . 1 = A   and     1 + B = 1

 

The proof is shown in Table below , which shows the truth table and the resulting logic circuit simplification.

 

  • Simplification Of Boolean Expression Using Boolean Algebra

 

  1. A simplified Boolean expression uses the fewest gates possible to implement a given expression.

 

Using Boolean algebra techniques, simplify this expression: AB + A(B + C) + B(B + C)

 

Solution

AB + A(B + C) + B(B + C)