Digital Fundamentals

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

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Digital FundamentalsDigital Fundamentals

CHAPTER 4 CHAPTER 4 Boolean Algebra and Logic SimplificationBoolean Algebra and Logic Simplification




Boolean Operations and Expressions Boolean Operations and Expressions




Boolean Operations and ExpressionsBoolean Operations and Expressions

• AdditionAddition0 + 0 = 00 + 0 = 00 + 1 = 10 + 1 = 11 + 0 = 11 + 0 = 11 + 1 = 11 + 1 = 1

• MultiplicationMultiplication0 * 0 = 00 * 0 = 00 * 1 = 00 * 1 = 01 * 0 = 01 * 0 = 01 * 1 = 11 * 1 = 1




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




Laws Boolean Algebra Laws Boolean Algebra

• Commutative LawsCommutative Laws• Associative LawsAssociative Laws• Distributive LawDistributive Law




Laws of Boolean AlgebraLaws of Boolean Algebra

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





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





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





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





• Distributive Law:Distributive Law:A(B + C) = AB + ACA(B + C) = AB + AC




Rules of Boolean AlgebraRules of Boolean Algebra





• Rule 1Rule 1

OR Truth Table





• Rule 2Rule 2

OR Truth Table





• Rule 3Rule 3

AND Truth Table





• Rule 4Rule 4

AND Truth Table





• Rule 5Rule 5

OR Truth Table




Rules of Boolean Algebra Rules of Boolean Algebra

• Rule 6Rule 6

OR Truth Table





• Rule 7Rule 7

AND Truth Table





• Rule 8Rule 8

AND Truth Table





• Rule 9Rule 9





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

AND Truth Table OR Truth Table





• Rule 11:Rule 11: BABAA






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





DeMorgan’s Theorem DeMorgan’s Theorem




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”




Standard Forms of Boolean ExpressionsStandard Forms of Boolean Expressions




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)




The Karnaugh Map The Karnaugh Map




The Karnaugh MapThe Karnaugh Map

3-Variable Karnaugh Map3-Variable Example





4-Variable Karnaugh Map

4-Variable Example





5-Variable Karnaugh Mapping




VHDLVHDL




VHDLVHDL

• VHDL OperatorsVHDL Operators

andandorornotnotnandnandnornor

xorxorxnorxnor

• VHDL ElementsVHDL Elements

entityentityarchitecturearchitecture




VHDLVHDL

• Entity StructureEntity Structure

Example:Example:

entityentity AND_Gate1 AND_Gate1 isisport(port(A,BA,B:in bit::in bit:XX:out bit);:out bit);

end entity end entity AND_Gate1AND_Gate1




VHDLVHDL

• ArchitectureArchitecture

Example:Example:

architecturearchitecture LogicFunction of AND_Gate1 LogicFunction of AND_Gate1 isisbeginbegin

XX<=<=A A andand B B;;end architecture end architecture LogicFunctionLogicFunction




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;;

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