1. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 1 Digital Fundamentals CHAPTER 4 Boolean Algebra and Logic Simplification

2. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 2 Boolean Operations and Expressions

3. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 3 Boolean Operations and Expressions AdditionAddition 0 + 0 = 0 0 + 1 = 1 1 + 0 = 1 1 + 1 = 1 MultiplicationMultiplication 0 * 0 = 0 0 * 1 = 0 1 * 0 = 0 1 * 1 = 1

4. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 4 Laws and Rules of Boolean Algebra Laws and Rules of Boolean Algebra

5. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 5 Laws Boolean Algebra Laws Boolean Algebra Commutative LawsCommutative Laws Associative LawsAssociative Laws Distributive LawDistributive Law

6. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 6 Laws of Boolean Algebra Commutative Law of Addition:Commutative Law of Addition: A + B = B + A

7. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 7 Laws of Boolean Algebra Commutative Law of Multiplication:Commutative Law of Multiplication: A * B = B * A

8. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 8 Laws of Boolean Algebra Laws of Boolean Algebra Associative Law of Addition:Associative Law of Addition: A + (B + C) = (A + B) + C

9. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 9 Laws of Boolean Algebra Laws of Boolean Algebra Associative Law of Multiplication:Associative Law of Multiplication: A * (B * C) = (A * B) * C

10. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 10 Laws of Boolean Algebra Laws of Boolean Algebra Distributive Law:Distributive Law: A(B + C) = AB + AC

11. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 11 Rules of Boolean Algebra

12. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 12 Rules of Boolean Algebra Rule 1Rule 1 OR Truth Table

13. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 13 Rules of Boolean Algebra Rule 2Rule 2 OR Truth Table

14. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 14 Rules of Boolean Algebra Rule 3Rule 3 AND Truth Table

15. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 15 Rules of Boolean Algebra Rule 4Rule 4 AND Truth Table

16. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 16 Rules of Boolean Algebra Rule 5Rule 5 OR Truth Table

17. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 17 Rules of Boolean Algebra Rule 6Rule 6 OR Truth Table

18. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 18 Rules of Boolean Algebra Rule 7Rule 7 AND Truth Table

19. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 19 Rules of Boolean Algebra Rule 8Rule 8 AND Truth Table

20. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 20 Rules of Boolean Algebra Rule 9Rule 9

21. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 21 Rules of Boolean Algebra Rule 10: A + AB = ARule 10: A + AB = A AND Truth TableOR Truth Table

22. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 22 Rules of Boolean Algebra Rule 11:Rule 11: AND Truth TableOR Truth Table

23. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 23 Rules of Boolean Algebra Rule 12: (A + B)(A + C) = A + BCRule 12: (A + B)(A + C) = A + BC AND Truth TableOR Truth Table

24. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 24 DeMorgans Theorem

25. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 25 DeMorgans Theorems Theorem 1Theorem 1 Theorem 2Theorem 2 Remember: Break the bar, change the sign

26. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 26 Standard Forms of Boolean Expressions

27. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 27 Standard Forms of Boolean Expressions The sum-of-product (SOP) formThe sum-of-product (SOP) form Example: X = AB + CD + EF The product of sum (POS) formThe product of sum (POS) form Example: X = (A + B)(C + D)(E + F)

28. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 28 The Karnaugh Map

29. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 29 The Karnaugh Map 3-Variable Karnaugh Map3-Variable Example

30. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 30 The Karnaugh Map 4-Variable Karnaugh Map 4-Variable Example

31. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 31 The Karnaugh Map 5-Variable Karnaugh Mapping

32. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 32 VHDL

33. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 33 VHDL VHDL OperatorsVHDL Operatorsandornotnandnorxorxnor VHDL ElementsVHDL Elementsentityarchitecture

34. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 34 VHDL VHDL Entity StructureEntity StructureExample: entity AND_Gate1 is port(A,B:in bit:X:out bit); end entity AND_Gate1

35. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 35 VHDL VHDL ArchitectureArchitectureExample: architecture LogicFunction of AND_Gate1 is begin X<=A and B; end architecture LogicFunction

36. Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New Jersey 07458 All rights reserved. Slide 36 Hardware Description Languages (HDL) Hardware Description Languages (HDL) Boolean Expressions in VHDLBoolean Expressions in VHDL ANDX <= A and B; ORX <= A or B; NOTX <= A not B; NANDX <= A nand B; NORX <= A nor B; XORX <= A xor B; XNORX <= A xnor B;