Slide 1 Digital Fundamentals CHAPTER 4 Boolean Algebra and Logic Simplification.
Boolean Algebra and Logic Simplification - - Get a
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1 Boolean Algebra and Logic Simplification
Transcript of Boolean Algebra and Logic Simplification - - Get a
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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
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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
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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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