Chapter 9
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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

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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


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:


ExamplePlease 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.


Example4Transform (PQ)R to CNF.

Ans.


9.3DNF Normal Form and Boolean Function


Example DNF (Disjunctive Normal Form)

Ans.

e. g.


ExampleTransform proposition logic to DNF.

Ans.

Four useful rules:


Example Transform PQ to DNF.


ExampleMap 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


Three Main Logic Gates


Example1LogicDesign for Full Adder.

Ans.

represents carry.


We have


Fig.9.4.2 Basic module for two-bit addition.


Two-bit adder module

Extension:

Fig.9.4.4 Logic design of X+Y


ExampleGray 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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