floyd digital fundamentals, 9/e copyright ©2006 by pearson education, inc. upper saddle river, new...

FloydFloydDigital Fundamentals, 9/eDigital Fundamentals, 9/e

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 = 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

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

YXYX Remember: Remember:

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

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

• VHDL OperatorsVHDL Operators

andand

notnot

nandnand

nornor

xorxor

xnorxnor

• VHDL ElementsVHDL Elements

entityentity

architecturearchitecture

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

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

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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