Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New...

FloydFloydDigital Fundamentals, 9/eDigital Fundamentals, 9/e

Copyright ©2006 by Pearson Education, Inc.Copyright ©2006 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458Upper Saddle River, New Jersey 07458

All rights reserved.All rights reserved.

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Digital FundamentalsDigital Fundamentals

CHAPTER 4 CHAPTER 4

Boolean Algebra and Logic SimplificationBoolean Algebra and Logic Simplification




Slide 2

Boolean Operations and Expressions Boolean Operations and Expressions




Slide 3

Boolean Operations and ExpressionsBoolean Operations and Expressions

• AdditionAddition0 + 0 = 00 + 0 = 0

0 + 1 = 10 + 1 = 1

1 + 0 = 11 + 0 = 1

1 + 1 = 11 + 1 = 1

• MultiplicationMultiplication0 * 0 = 00 * 0 = 0

0 * 1 = 00 * 1 = 0

1 * 0 = 01 * 0 = 0

1 * 1 = 11 * 1 = 1




Slide 4

Laws and Rules of Boolean Algebra Laws and Rules of Boolean Algebra




Slide 5

Laws Boolean Algebra Laws Boolean Algebra

• Commutative LawsCommutative Laws

• Associative LawsAssociative Laws

• Distributive LawDistributive Law




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Laws of Boolean AlgebraLaws of Boolean Algebra

• Commutative Law of Addition:Commutative Law of Addition:A + B = B + AA + B = B + A




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• Commutative Law of Multiplication:Commutative Law of Multiplication:A * B = B * AA * B = B * A




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• Associative Law of Addition:Associative Law of Addition:A + (B + C) = (A + B) + CA + (B + C) = (A + B) + C




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• Associative Law of Multiplication:Associative Law of Multiplication:A * (B * C) = (A * B) * CA * (B * C) = (A * B) * C




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• Distributive Law:Distributive Law:

A(B + C) = AB + ACA(B + C) = AB + AC




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Rules of Boolean AlgebraRules of Boolean Algebra




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• Rule 1Rule 1

OR Truth Table




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• Rule 2Rule 2

OR Truth Table




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• Rule 3Rule 3

AND Truth Table




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• Rule 4Rule 4

AND Truth Table




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• Rule 5Rule 5

OR Truth Table




Slide 17

Rules of Boolean Algebra Rules of Boolean Algebra

• Rule 6Rule 6

OR Truth Table




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• Rule 7Rule 7

AND Truth Table




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• Rule 8Rule 8

AND Truth Table




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• Rule 9Rule 9




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• Rule 10: A + AB = ARule 10: A + AB = A

AND Truth Table OR Truth Table




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• Rule 11:Rule 11: BABAA





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• Rule 12: (A + B)(A + C) = A + BCRule 12: (A + B)(A + C) = A + BC





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DeMorgan’s Theorem DeMorgan’s Theorem




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DeMorgan’s TheoremsDeMorgan’s Theorems

• Theorem 1Theorem 1

• Theorem 2Theorem 2

YXXY

YXYX Remember: Remember:

““Break the bar, Break the bar, change the sign”change the sign”




Slide 26

Standard Forms of Boolean ExpressionsStandard Forms of Boolean Expressions




Slide 27

Standard Forms of Boolean ExpressionsStandard Forms of Boolean Expressions

• The sum-of-product (SOP) formThe sum-of-product (SOP) formExample: X = AB + CD + EFExample: X = AB + CD + EF

• The product of sum (POS) formThe product of sum (POS) formExample: X = (A + B)(C + D)(E + F)Example: X = (A + B)(C + D)(E + F)




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The Karnaugh Map The Karnaugh Map




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The Karnaugh MapThe Karnaugh Map

3-Variable Karnaugh Map3-Variable Example




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4-Variable Karnaugh Map

4-Variable Example




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5-Variable Karnaugh Mapping




Slide 32

VHDLVHDL




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VHDLVHDL

• VHDL OperatorsVHDL Operators

andand

oror

notnot

nandnand

nornor

xorxor

xnorxnor

• VHDL ElementsVHDL Elements

entityentity

architecturearchitecture




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VHDLVHDL

• Entity StructureEntity Structure

Example:Example:

entityentity AND_Gate1 AND_Gate1 isis

port(port(A,BA,B:in bit::in bit:XX:out bit);:out bit);

end entity end entity AND_Gate1AND_Gate1




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VHDLVHDL

• ArchitectureArchitecture

Example:Example:

architecturearchitecture LogicFunction of AND_Gate1 LogicFunction of AND_Gate1 isis

beginbegin

XX<=<=A A andand B B;;

end architecture end architecture LogicFunctionLogicFunction




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Hardware Description Languages (HDL)Hardware Description Languages (HDL)

• Boolean Expressions in VHDLBoolean Expressions in VHDLANDAND X X <=<= A A andand B B;;

OROR X X <=<= A A oror B B;;

NOTNOT X X <=<= A A notnot B B;;

NANDNAND X X <=<= A A nandnand B B;;

NORNOR X X <=<= A A nornor B B;;

XORXOR X X <=<= A A xorxor B B;;

XNORXNOR X X <=<= A A xnorxnor B B;;

Floyd Digital Fundamentals, 9/e Copyright ©2006 by Pearson Education, Inc. Upper Saddle River, New...

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