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Digital Fundamentals, Eleventh EditionThomas L. Floyd
Copyright © 2015 by Pearson Education, Inc.All Rights Reserved
Digital Fundamentals
Boolean Algebra and Logic Simplification
Digital Fundamentals, Eleventh EditionThomas L. Floyd
Copyright © 2015 by Pearson Education, Inc.All Rights Reserved
FIGURE 4-1
Digital Fundamentals, Eleventh EditionThomas L. Floyd
Copyright © 2015 by Pearson Education, Inc.All Rights Reserved
FIGURE 4-2
Digital Fundamentals, Eleventh EditionThomas L. Floyd
Copyright © 2015 by Pearson Education, Inc.All Rights Reserved
FIGURE Application of commutative law of addition.
Digital Fundamentals, Eleventh EditionThomas L. Floyd
Copyright © 2015 by Pearson Education, Inc.All Rights Reserved
FIGURE Application of commutative law of multiplication.
Digital Fundamentals, Eleventh EditionThomas L. Floyd
Copyright © 2015 by Pearson Education, Inc.All Rights Reserved
FIGURE Application of associative law of addition.
Digital Fundamentals, Eleventh EditionThomas L. Floyd
Copyright © 2015 by Pearson Education, Inc.All Rights Reserved
FIGURE Application of associative law of multiplication.
Digital Fundamentals, Eleventh EditionThomas L. Floyd
Copyright © 2015 by Pearson Education, Inc.All Rights Reserved
FIGURE Application of distributive law.
Digital Fundamentals, Eleventh EditionThomas L. Floyd
Copyright © 2015 by Pearson Education, Inc.All Rights Reserved
LAWS
• commutative law: A+B =?
AB= ?
• associative law: A +(B+C) =?
A(BC) =?
• distributive law: A(B+C)=?
(A+B)(C+D)=?
Digital Fundamentals, Eleventh EditionThomas L. Floyd
Copyright © 2015 by Pearson Education, Inc.All Rights Reserved
TABLE 4–1 Basic rules of Boolean algebra.
Digital Fundamentals, Eleventh EditionThomas L. Floyd
Copyright © 2015 by Pearson Education, Inc.All Rights Reserved
FIGURE 4-8
Anything ORed with 0 equals to itself
Digital Fundamentals, Eleventh EditionThomas L. Floyd
Copyright © 2015 by Pearson Education, Inc.All Rights Reserved
FIGURE 4-9
Anything ORed with 1 equals to 1
Digital Fundamentals, Eleventh EditionThomas L. Floyd
Copyright © 2015 by Pearson Education, Inc.All Rights Reserved
FIGURE 4-10
Anything ANDed with 0 equals to 0
Digital Fundamentals, Eleventh EditionThomas L. Floyd
Copyright © 2015 by Pearson Education, Inc.All Rights Reserved
FIGURE 4-11
Anything ANDed with 1 equals to
itself
Digital Fundamentals, Eleventh EditionThomas L. Floyd
Copyright © 2015 by Pearson Education, Inc.All Rights Reserved
FIGURE 4-12
Anything ORed with itself equals to
itself
Digital Fundamentals, Eleventh EditionThomas L. Floyd
Copyright © 2015 by Pearson Education, Inc.All Rights Reserved
FIGURE 4-13
Anything ORed with its own
complement equals to 1
Digital Fundamentals, Eleventh EditionThomas L. Floyd
Copyright © 2015 by Pearson Education, Inc.All Rights Reserved
FIGURE 4-14
Anything ANDed with its own
complement equals to itself
Digital Fundamentals, Eleventh EditionThomas L. Floyd
Copyright © 2015 by Pearson Education, Inc.All Rights Reserved
FIGURE 4-15
Anything ANDed with its own
complement equals to 0
Digital Fundamentals, Eleventh EditionThomas L. Floyd
Copyright © 2015 by Pearson Education, Inc.All Rights Reserved
FIGURE 4-16
Anything complemented twice will
return to its original logic level
Digital Fundamentals, Eleventh EditionThomas L. Floyd
Copyright © 2015 by Pearson Education, Inc.All Rights Reserved
TABLE 4–2 Rule 10: A + AB = A.
Elimination: One or more variables in
second term can be eliminated.
Used in simplication
Digital Fundamentals, Eleventh EditionThomas L. Floyd
Copyright © 2015 by Pearson Education, Inc.All Rights Reserved
TABLE 4–3 Rule 11: A + ĀB = A + B.
Absorption: One or more variables in second term can be
eliminated. Used in simplication
Ā+AB =Ā+B (?)
Digital Fundamentals, Eleventh EditionThomas L. Floyd
Copyright © 2015 by Pearson Education, Inc.All Rights Reserved
TABLE 4–4 Rule 12: (A + B)(A + C) = A + BC.
Absorption: Used in simplication
Digital Fundamentals, Eleventh EditionThomas L. Floyd
Copyright © 2015 by Pearson Education, Inc.All Rights Reserved
DEMORGAN'S THEOREMS
Digital Fundamentals, Eleventh EditionThomas L. Floyd
Copyright © 2015 by Pearson Education, Inc.All Rights Reserved
DEMORGAN'S THEOREMS
Digital Fundamentals, Eleventh EditionThomas L. Floyd
Copyright © 2015 by Pearson Education, Inc.All Rights Reserved
FIGURE 4-17 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.
Digital Fundamentals, Eleventh EditionThomas L. Floyd
Copyright © 2015 by Pearson Education, Inc.All Rights Reserved
FIGURE 4-18 A combinational logic circuit showing the development of the Boolean expression for the output.
Digital Fundamentals, Eleventh EditionThomas L. Floyd
Copyright © 2015 by Pearson Education, Inc.All Rights Reserved
FIGURE 4-20 Gate circuits
Digital Fundamentals, Eleventh EditionThomas L. Floyd
Copyright © 2015 by Pearson Education, Inc.All Rights Reserved
Simplify the following Boolean expressions :
STANDARD FORMS OF BOOLEAN EXPRESSIONS
Digital Fundamentals, Eleventh EditionThomas L. Floyd
Copyright © 2015 by Pearson Education, Inc.All Rights Reserved
FIGURE 4-22 Implementation of the SOP expression AB + BCD + AC.
STANDARD FORMS OF BOOLEAN EXPRESSIONS
Digital Fundamentals, Eleventh EditionThomas L. Floyd
Copyright © 2015 by Pearson Education, Inc.All Rights Reserved
FIGURE 4-23 This NAND/NAND implementation is equivalent to the AND/OR in Figure 4–22.
STANDARD FORMS OF BOOLEAN EXPRESSIONS
Digital Fundamentals, Eleventh EditionThomas L. Floyd
Copyright © 2015 by Pearson Education, Inc.All Rights Reserved
FIGURE 4-24 Implementation of the POS expression (A + B)(B + C + D)(A + C).
STANDARD FORMS OF BOOLEAN EXPRESSIONS
Digital Fundamentals, Eleventh EditionThomas L. Floyd
Copyright © 2015 by Pearson Education, Inc.All Rights Reserved
Circuit Simplification Using Boolean Algebra
Digital Fundamentals, Eleventh EditionThomas L. Floyd
Copyright © 2015 by Pearson Education, Inc.All Rights Reserved
Circuit Simplification Using Boolean Algebra
Digital Fundamentals, Eleventh EditionThomas L. Floyd
Copyright © 2015 by Pearson Education, Inc.All Rights Reserved
Karnaugh Map
• K-Map: a tools to preform a systematic reduction of complex logic circuits into simplified equivalent circuit.
• Transform the Boolean equation to be reduced into SOP expression
• Fill in a proper cell of the K-map
• Encircle adjacent cells in groups of two, four, or eight
• Find each term of the final SOP equation by determining which variable remain constant within each circle.
Digital Fundamentals, Eleventh EditionThomas L. Floyd
Copyright © 2015 by Pearson Education, Inc.All Rights Reserved
FIGURE 4-25 A 3-variable Karnaugh map showing Boolean product terms for each cell.
Digital Fundamentals, Eleventh EditionThomas L. Floyd
Copyright © 2015 by Pearson Education, Inc.All Rights Reserved
FIGURE 4-26 A 4-variable Karnaugh map.
Digital Fundamentals, Eleventh EditionThomas L. Floyd
Copyright © 2015 by Pearson Education, Inc.All Rights Reserved
FIGURE 4–27 Adjacent cells on a Karnaugh map are those that differ by only one variable. Arrows point between adjacent cells.
Digital Fundamentals, Eleventh EditionThomas L. Floyd
Copyright © 2015 by Pearson Education, Inc.All Rights Reserved
FIGURE 4-28 Example of mapping a standard SOP expression.
Transform the Boolean equation to be reduced into SOP expression
Fill in a proper cell of the K-map
Digital Fundamentals, Eleventh EditionThomas L. Floyd
Copyright © 2015 by Pearson Education, Inc.All Rights Reserved
FIGURE 4-29
Digital Fundamentals, Eleventh EditionThomas L. Floyd
Copyright © 2015 by Pearson Education, Inc.All Rights Reserved
FIGURE 4-30
Digital Fundamentals, Eleventh EditionThomas L. Floyd
Copyright © 2015 by Pearson Education, Inc.All Rights Reserved
FIGURE 4-33
Encircle adjacent cells in groups of two, four, or eight ?
Digital Fundamentals, Eleventh EditionThomas L. Floyd
Copyright © 2015 by Pearson Education, Inc.All Rights Reserved
FIGURE 4-34
Encircle adjacent cells in groups of two, four, or eight
Digital Fundamentals, Eleventh EditionThomas L. Floyd
Copyright © 2015 by Pearson Education, Inc.All Rights Reserved
FIGURE 4-35
Find each term of the final SOP equation by determining which
variable remain constant within each circle.
Digital Fundamentals, Eleventh EditionThomas L. Floyd
Copyright © 2015 by Pearson Education, Inc.All Rights Reserved
FIGURE 4-36
Find each term of the final SOP equation by determining
which variable remain constant within each circle.
Digital Fundamentals, Eleventh EditionThomas L. Floyd
Copyright © 2015 by Pearson Education, Inc.All Rights Reserved
FIGURE 4-37
Digital Fundamentals, Eleventh EditionThomas L. Floyd
Copyright © 2015 by Pearson Education, Inc.All Rights Reserved
FIGURE 4-39 Example of mapping directly from a truth table to a Karnaugh map.
Digital Fundamentals, Eleventh EditionThomas L. Floyd
Copyright © 2015 by Pearson Education, Inc.All Rights Reserved
FIGURE 4-43 Example of mapping a standard POS expression.