Boolean Algebra and Logic Simplification. Boolean Addition & Multiplication Boolean Addition...

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Boolean Algebra and Logic Simplification

Transcript of Boolean Algebra and Logic Simplification. Boolean Addition & Multiplication Boolean Addition...

Boolean Algebra and Logic Simplification

Boolean Addition & Multiplication

• Boolean Addition performed by OR gate• Sum Term describes Boolean Addition

• Boolean Multiplication performed by AND gate• Product Term describes Boolean Multiplication

CBA

Boolean Addition

• Sum of literals

• Sum term = 1 if any literal = 1• Sum term = 0 if all literals = 0

BA BA CBA

Boolean Multiplication

• Product of literals

• Product term = 1 if all literals = 1• Product term = 0 if any one literal = 0

BA. BA. CBA ..

Laws, Rules & Theorems of Boolean Algebra

• Commutative Law for addition and multiplication

• Associative Lawfor addition and multiplication

• Distributive Law• Rules of Boolean Algebra• Demorgan’s Theorems

Commutative Law

• Commutative Law for AdditionA + B = B + A

• Commutative Law for MultiplicationA.B = B.A

A

B

A + B A + BB

A

A

B

A.B A.BB

A

Associative Law

• Associative Law for Addition A + (B + C) = (A + B) + C

A

B

A+(B+C)

C

B+C

A

B

C

A+B

(A+B)+C

Associative Law

• Associative Law for MultiplicationA.(B.C) = (A.B).C

A

B

A.(B.C)

C

B.C

A

B

C

A.B

(A.B).C

Distributive Law

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

A

B

A.(B+C)

C

B+C

A

B

C

A.B

A.B+A.CA

A.C

Rules of Boolean Algebra

1. A + 0 = A2. A + 1 = 13. A.0 = 04. A.1 = A5. A + A = A6. A + = 1

7. A.A = A8. A. = 09. = A10. A + A.B = A11. A + = A + B12. (A+B).(A+C)

= A+B.C

A

A

A

BA.

Demorgan’s Theorems

• First Theorem

• Second Theorem

BABA .

BABA .

A

BB.A

A

BBA

A

BBA

A

BB.A

Demorgan’s Theorems

• Any number of variables

• Combination of variables

ZYXZYX ..

ZYXZYX ..

).().().).(.( BCACBABCACBA

BCACBA .).().(. BCACBA ).().(

CBBACABA ....

CBCABA ...

Simplification using Boolean Algebra

• AB + A(B+C) + B(B+C)= AB + AB + AC + BB +BC= AB + AC + B + BC= AB + AC + B= B + AC

Simplified Circuit

A

B

C

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

A

B

C

B+AC

Some More Examples

• Explanation with rule is given

Do your self

Standard forms of Boolean Expressions

Standardization make the implementation of Boolean expression much more easier and systematic

Two forms are• Sum-of-Products form• Product-of-Sums form

Standard forms of Boolean Expressions

• Sum-of-Products formAB + ABCABC + CDE +

• Product-of-Sums form

DCBACCBABA

))(( CBABA

))()(( DCBEDCCBA

))()(( CACBABA

Implementation of SOP expression

AB

C

B+AC+AD

A

D

Implementation of POS expression

D

BC

(A+B)(B+C+D)(A+C)

AB

AC

Conversion of general expression to SOP form

BADAC

BEFBCDABEFCDBAB )(

BDBCBADACABDCBBA ))((

CBCACBACBACBA )()()(