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Chapter 9. Normal Forms and Logic Design. 9.2 PNF and CNF Normal Forms 9.3 DNF Normal Form and Boolean Function 9.4 Logic Design PNF:Prenix Normal Form CNF:Conjunction Normal Form DNF:Disjunctive Normal Form. 9.2 PNF and CNF Normal Forms. Example PNF

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Chapter 9

Chapter 9

Normal Formsand Logic Design


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


9 2 pnf and cnf normal forms
9.2 PNF and CNF Normal Forms


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:


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


Example CNF(Conjunction Normal Form)

Ans.

ci:clause

pij:literal

e. g.


Example4 Transform (PQ)R to CNF.

Ans.


9 3 dnf normal form and boolean function
9.3DNF Normal Form and Boolean Function


Example DNF (Disjunctive Normal Form)

Ans.

e. g.


Example Transform proposition logic to DNF.

Ans.

Four useful rules:


Example Transform PQ to DNF.


Example Map Table to DNF

Ans.


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


9 4 logic design
9.4 Logic Design



Example1 LogicDesign for Full Adder.

Ans.

represents carry.



Fig.9.4.2 Basic module for two-bit addition.


Two-bit adder module

Extension:

Fig.9.4.4 Logic design of X+Y


Example Gray code.

Ans.

Also called Reflected Code

Two-bit Gray code:

0 0

0 1

1 1

1 0


Three-bit Gray code:

Mirror

0 0

0 1

1 1

1 0

U

1 0

1 1

0 1

0 0

L


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


ExampleInteger to Gray code.

e.g.

b=(01)2, we have g=(g1g0)=(01)


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