INTRO TO I & CT. (Boolean Algebra and Logic Simplification.) Lecture # By: Department of CS & IT.

Transcription

1 INTRO TO I & CT. (Boolean Algebra and Logic Simplification.) Lecture # By: M.Nadeem Akhtar. Department of CS & IT. URL: 1

2 Boolean Algebra George Boole lived in England during the time Abraham Lincoln was getting involved in politics in the United States. Boole was a mathematician and logician who developed ways of expressing logical processes using algebraic symbols, thus creating a branch of mathematics known as symbolic logic, or Boolean algebra. Today, all our computers use Boole's logic system - using microchips that contain thousands of tiny electronic switches arranged into logical gates.

3 Boolean Algebra Boolean algebra is an algebra for the manipulation of objects that can take on only two values, typically true and false, although it canbeanypairofvalues. Because computers are built as collections of switches that are either on or off, Boolean algebra is a very natural way to represent digital information.

4 In reality, digital circuits use low and high voltages, but for our level of understanding, 0 and 1 will sufficient. Itis common to interpretthe digitalvalue 0as falseandthedigitalvalue1astrue.

5 A + B + C 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

6 Sum of literals Boolean Addition A + B A + B A + B + Sum term = 1 if any literal = 1 Sum term = 0 if all literals = 0 C

7 Product of literals Boolean Multiplication A. B A. B A. B. C Product term = 1 if all literals = 1 Product term = 0 if any one literal = 0

8 Laws, Rules & Theorems of Boolean Algebra Commutative Law for addition and multiplication Associative Law for addition and multiplication Distributive Law Rules of Boolean Algebra Demorgan s Theorems

9 Commutative Law Commutative Law for Addition A + B = B + A Commutative Law for Multiplication A.B = B.A

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

11 Associative Law Associative Law for Multiplication A.(B.C) = (A.B).C

12 A.(B + C) = A.B + A.C Distributive Law

13 Rules of Boolean Algebra 1. A + 0 = A 2. A + 1 = 1 3. A.0 = 0 4. A.1 = A 5. A + A = A 6. A + A= 1 7. A.A = A 8. A. A= 0 9. = A A 10.A + A.B = A 11.A + = A + B A.B 12.(A+B).(A+C) = A+B.C

14 First Theorem Demorgan s Theorems A. B = A + B A.B A + B Second Theorem A + B = A. B A + B A. B

15 Demorgan s Theorems Any number of variables X. Y. Z = X + Y + Z X + Y + Z = X. Y. Z Combination of variables ( A + B. C).( A. C + B) = ( A + B. C) + ( A. C + B) = A.( B. C) + ( A. C). B = A.( B + C) + ( A + C). B = = A. B + A. C + A. B + B. C A. B + A. C + B. C

16 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

17 Simplified Circuit

18 Simplify C+ BC C+BC = C+( B + C ) = ( C+ C )+ B = 1+B = 1

19 Simplify AB( A+B)(B + B) AB( A+B)(B + B) = AB( A+B) = (A +B)( A+B) = A A+AB+B A+BB = A+AB+B A = A+A(B+B ) = A +A = A

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21 Simplify A( A+B)+(B + AA)(A + B ) A( A+B)+(B + AA)(A + B ) = AA + AB +(B+ A) (A+B) = AB + BA + BB +AA+AB = AB + BA+ A+AB =B(A+A) +A(1+B) = B+A

22 Some More Examples Do your self

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26 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

27 Standard forms of Boolean Sum-of-Products form AB + ABC ABC + CDE + Expressions BCD A B + ABC + AC Product-of-Sums form ( A + B)( A + B + C) ( A + B + C)( C + D + E)( B + C + D) ( A + B)( A + B + C)( A + C)

28 Implementation of SOP expression

29 Implementation of POS expression

30 Conversion of general expression to SOP form AB + B( CD + EF) = AB + BCD + BEF ( A + B)( B + C + D) = AB + AC + AD + B + BC + BD = AC + AD + B ( A + B) + C = ( A + B) C = ( A + B) C = AC + BC