หน่วยที่ 4 บูลลีนและการลดรูปสมการ

Boolean Algebra and Logic Simplification 1

BOOLEAN OPERATIONS AND EXPRESSIONS ï‚¢

Boolean Addition (OR)

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ï‚¢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. 7

APPLICATION OF COMMUTATIVE LAW OF MULTIPLICATION. 8

APPLICATION OF ASSOCIATIVE LAW OF ADDITION 9

APPLICATION OF ASSOCIATIVE LAW OF MULTIPLICATION 10

APPLICATION OF DISTRIBUTIVE LAW 11

RULES OF BOOLEAN ALGEBRA

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RULE 1 13

RULE 2 14

RULE 3 15

RULE 4

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RULE 5 17

RULE 6 18

RULE 7 19

RULE 8 20

RULE 9

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DEMORGAN’S THEOREM

Gate equivalencies and the corresponding truth tables that illustrate 25 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

A logic circuit showing the development of the Boolean expression for the output.

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CONSTRUCTING A TRUTH TABLE FOR A LOGIC CIRCUIT ï‚¢ 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 31

STANDARD FORMS OF BOOLEAN EXPRESSIONS  Sum-of-Products

(SOP) Form

• Product-of-Sum (POS) Form

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SOP FORM

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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POS Form

IMPLEMENTATION OF THE POS EXPRESSION (A + B)(B + C + D) (A + C).

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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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Karnaugh Map

A 3-VARIABLE KARNAUGH

MAP SHOWING PRODUCT TERMS.

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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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Karnaugh Map SOP Minimization

EXAMPLE OF MAPPING

A STANDARD

SOP EXPRESSION.

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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

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. 59

101+011+001+000+100

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MAPPING DIRECTLY FROM TRUTH TABLE

Example of mapping directly from a truth table to a Karnaugh map.

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“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

Example of mapping a standard POS expression.

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Example 4-30

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CONVERTING B/W POS AND SOP USING KARNAUGH MAP

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FIVE-VARIABLE KARNAUGH MAPS

A 5-variable Karnaugh map.

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ILLUSTRATION OF GROUPINGS OF 1S IN ADJACENT CELLS OF A 5-VARIABLE MAP.

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Programmable Logic

BASIC STRUCTURE OF A PAL.

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PAL IMPLEMENTATION OF A SUM-OFPRODUCTS EXPRESSION.

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Digital System Application

SEVEN-SEGMENT DISPLAY FORMAT SHOWING ARRANGEMENT OF SEGMENTS.

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DISPLAY OF DECIMAL DIGITS WITH A 7-SEGMENT DEVICE. 77

ARRANGEMENTS OF 7-SEGMENT LED DISPLAYS.

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BLOCK DIAGRAM OF 7-SEGMENT LOGIC AND DISPLAY.

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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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