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Understanding Normal Forms in Logic Design: PNF, CNF, and DNF Explained

This chapter delves into the essential concepts of normal forms in logic design, highlighting Prenex Normal Form (PNF), Conjunctive Normal Form (CNF), and Disjunctive Normal Form (DNF). Detailed examples demonstrate the transformation of logical propositions into these forms, along with practical applications in Boolean functions. Additionally, the chapter touches upon important rules such as Idempotent Law and Demorgan's Law, and provides insights into fundamental logic gates, including designs for adders and Gray code conversions.

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Understanding Normal Forms in Logic Design: PNF, CNF, and DNF Explained

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  1. Chapter 9 Normal Formsand Logic Design

  2. 9.2PNF and CNF Normal Forms 9.3DNF Normal Form and Boolean Function 9.4Logic Design PNF:Prenix Normal Form CNF:Conjunction Normal Form DNF:Disjunctive Normal Form

  3. 9.2 PNF and CNF Normal Forms

  4. Example PNF (1) x in P(x) and x in Q(x) are in different domains, i.e. two x’s are different local variable transform it to the following PNF:

  5. Example Please transform x y ((z (P(x, z) Q(y, z))r R(x, y, r)) to PNF. Ans. It can be transformed to

  6. Example CNF(Conjunction Normal Form) Ans. ci:clause pij:literal e. g.

  7. Example4 Transform (PQ)R to CNF. Ans.

  8. 9.3DNF Normal Form and Boolean Function

  9. Example DNF (Disjunctive Normal Form) Ans. e. g.

  10. Example Transform proposition logic to DNF. Ans. Four useful rules:

  11. Example Transform PQ to DNF.

  12. Example Map Table to DNF Ans.

  13. Rule (2) is called Idempotent Law. Rule (3) is called Distributive Law. Rule (4) is called Demorgan Law.

  14. 9.4 Logic Design

  15. Three Main Logic Gates

  16. Example1 LogicDesign for Full Adder. Ans. represents carry.

  17. We have

  18. Fig.9.4.2 Basic module for two-bit addition.

  19. Two-bit adder module Extension: Fig.9.4.4 Logic design of X+Y

  20. Example Gray code. Ans. Also called Reflected Code Two-bit Gray code: 0 0 0 1 1 1 1 0

  21. Three-bit Gray code: Mirror 0 0 0 1 1 1 1 0 U 1 0 1 1 0 1 0 0 L

  22. 0U and 1L: 0 0 0 0 0 1 0 1 1 0 1 0 1 1 0 1 1 1 1 0 1 1 0 0 0U 1L

  23. ExampleInteger to Gray code. … e.g. b=(01)2, we have g=(g1g0)=(01)

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