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Boolean Algebra

and

Logic Simplification

BOOLEAN OPERATIONS AND EXPRESSIONS

Boolean Addition (OR)

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Boolean Multiplication (AND)

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LAWS AND RULES OF

BOOLEAN ALGEBRA

Laws:

Commutative

Associative

Distributive

Rules

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APPLICATION OF COMMUTATIVE

LAW OF ADDITION.

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APPLICATION OF COMMUTATIVE LAW

OF MULTIPLICATION.

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APPLICATION OF ASSOCIATIVE

LAW OF ADDITION

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APPLICATION OF ASSOCIATIVE

LAW OF MULTIPLICATION

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APPLICATION OF

DISTRIBUTIVE LAW 11

RULES OF BOOLEAN ALGEBRA

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RULE 1

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RULE 2

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RULE 3

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RULE 4

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RULE 5

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RULE 6

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RULE 7

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RULE 8

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RULE 9

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DEMORGAN’S THEOREM

25 Gate equivalencies and the corresponding truth tables that illustrate

DeMorgan’s theorems. Notice the equality of the two output columns

in each table. This shows that the equivalent gates perform the same logic function.

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BOOLEAN EXPRESSION

FOR A LOGIC CIRCUIT

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A logic circuit showing the development

of the Boolean expression for the output.

CONSTRUCTING A TRUTH TABLE

FOR A LOGIC CIRCUIT

Evaluating the

expression and

putting results in

truth table format

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SIMPLIFICATION USING BOOLEAN ALGEBRA

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GATE CIRCUITS FOR EXAMPLE 4-8

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STANDARD FORMS

OF BOOLEAN EXPRESSIONS

Sum-of-Products (SOP) Form

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• Product-of-Sum (POS) Form

SOP FORM

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Implementation of the SOP expression

AB + BCD + AC.

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STANDARD SOP FORM

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BINARY REPRESENTATION

OF PRODUCT TERM

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IMPLEMENTATION OF THE POS EXPRESSION

(A + B)(B + C + D) (A + C). 38

POS Form

STANDARD POS FORM

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BINARY REPRESENTATION OF SUM TERM

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CONVERTING STANDARD SOP

TO STANDARD POS

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CONVERTING SOP TO TRUTH TABLE

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CONVERTING POS TO TRUTH TABLE

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DETERMINING STANDARD

EXPRESSIONS FROM TRUTH TABLE

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A 3-VARIABLE KARNAUGH MAP

SHOWING PRODUCT TERMS.

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Karnaugh Map

A 4-VARIABLE KARNAUGH MAP. 47

ADJACENT CELLS ON A KARNAUGH MAP ARE THOSE

THAT DIFFER BY ONLY ONE VARIABLE. ARROWS POINT

BETWEEN ADJACENT CELLS.

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EXAMPLE OF MAPPING A STANDARD SOP EXPRESSION. 49

Karnaugh Map SOP Minimization

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MAPPING NONSTANDARD

SOP EXPRESSION

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KARNOUGH MAP SIMPLIFICATION OF

SOP EXPRESSIONS

Example 4-25 Group the 1s in each

Karnaugh maps

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DETERMINE SOP

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Related Problem: add 1 in the lower right cell (1010) and

determine the resulting SOP.

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Related Problem: For the Karnaugh map in Fig. 4-31(d), add 1

in the 0111 cell and determine the resulting SOP.

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101+011+001+000+100

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MAPPING DIRECTLY FROM TRUTH TABLE

62 Example of mapping directly from

a truth table to a Karnaugh map.

“DON’T CARE” CONDITIONS

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Example of the use of “don’t care” conditions to simplify an expression.

KARNAUGH MAP POS

MINIMIZATION

64 Example of mapping a standard POS

expression.

Example 4-30

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CONVERTING B/W POS AND SOP USING KARNAUGH MAP

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FIVE-VARIABLE KARNAUGH MAPS

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A 5-variable Karnaugh map.

ILLUSTRATION OF GROUPINGS OF 1S

IN ADJACENT CELLS OF A 5-VARIABLE MAP.

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BASIC STRUCTURE OF A PAL. 74

Programmable Logic

PAL IMPLEMENTATION OF A SUM-OF-

PRODUCTS EXPRESSION. 75

SEVEN-SEGMENT DISPLAY FORMAT SHOWING

ARRANGEMENT OF SEGMENTS. 76

Digital System Application

DISPLAY OF DECIMAL DIGITS

WITH A 7-SEGMENT DEVICE.

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ARRANGEMENTS OF 7-SEGMENT LED DISPLAYS.

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BLOCK DIAGRAM OF 7-SEGMENT

LOGIC AND DISPLAY. 79

KARNAUGH MAP MINIMIZATION OF THE

SEGMENT-A LOGIC EXPRESSION.

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THE MINIMUM LOGIC IMPLEMENTATION FOR

SEGMENT A OF THE 7-SEGMENT DISPLAY.

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