Lecture #7 - Boolean Algebra and Standard forms of Boolean expressions(1)

Lecture #7 - Boolean Algebra and Standard forms of Boolean expressions(1)

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ECE 301 – Digital Electronics Boolean Algebra, Standard forms of Boolean Expressions (Lecture #7)

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Learning Objectives Boolean Algebra laws and theorems Simplifying Boolean expressions using Boolean algebra Standard forms of Boolean expressions Sum of Products (SOP) Product of Sums (POS) Spring 2012 ECE 301 - Digital 2
Reading Sections 2.4 – 2.8, 3.1 – 3.5 Spring 2012 ECE 301 - Digital 3

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Boolean Algebra Spring 2012 ECE 301 - Digital 4
Boolean Laws and Theorems Spring 2012 ECE 301 - Digital 5 Operations with 0 and 1: 1. X + 0 = X 1D. X • 1 = X 2. X + 1 = 1 2D. X • 0 = 0 Idempotent laws: 3. X + X = X 3D. X X = X Involution law: 4. ( X' ) ' = X Laws of complementarity:

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Boolean Laws and Theorems Spring 2012 ECE 301 - Digital 6 Commutative laws: 6. X + Y = Y + X 6D. XY = YX Associative laws: 7. ( X + Y ) + Z = X + ( Y + Z ) 7D. ( XY ) Z = X ( YZ ) = XYZ = X + Y + Z Distributive laws: 8. X ( Y + Z ) = XY + XZ 8D. X + YZ = (X + Y )( X + Z ) Simplification theorems: 9. XY + XY' = X 9D. ( X + Y )( X + Y' ) = X 10. X + XY = X 10D. X ( X + Y ) = X
Boolean Laws and Theorems Spring 2012 ECE 301 - Digital 7 DeMorgan's laws: 12. ( X + Y + Z +...) ' = X'Y'Z'. .. 12D. ( XYZ ...) ' = X' + Y' + Z' +... Duality: 13. ( X + Y + Z +...) D = XYZ ... 13D. ( XYZ ...) D = X + Y + Z +... Theorem for multiplying out and factoring: 14. ( X + Y )( X' + Z ) = XZ + X'Y 14D. XY + X'Z = ( X + Z )( X' + Y ) Consensus theorems: 15. XY + YZ + X'Z = XY + X'Z 15D. ( X + Y )( Y + Z )(X ' + Z ) = ( X + Y )( X' + Z )

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Boolean Laws and Theorems Boolean Laws and Theorems can be proven using: 1. Truth Tables 2. Boolean Algebra Spring 2012 ECE 301 - Digital 8
Distributive Law: Exercise #1 Use the distributive law to multiply out the following Boolean expression: F = (A + C).(B’ + D’) Spring 2012 ECE 301 - Digital 9 Distributive law (8): X.(Y + Z) = X.Y + X.Z

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Distributive Law: Exercise #2 Use the distributive law to factor the following Boolean expression: F = (A’.B) + (C.D’) Spring 2012 ECE 301 - Digital 10 Distributive law (8D): X + Y.Z = (X+Y).(X+Z)
Simplification Thm: Exercise #1 Use the simplification theorems

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This note was uploaded on 03/26/2012 for the course ECE 301 taught by Professor Staff during the Spring '08 term at George Mason.

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Lecture #7 - Boolean Algebra and Standard forms of Boolean expressions(1)

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