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ECE 301 – Digital Electronics

Boolean Algebra and Standard Forms of Boolean Expressions (Lecture #4). ECE 301 – Digital Electronics. The slides included herein were taken from the materials accompanying Fundamentals of Logic Design, 6 th Edition , by Roth and Kinney,

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ECE 301 – Digital Electronics

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  1. Boolean Algebra and Standard Forms of Boolean Expressions (Lecture #4) ECE 301 – Digital Electronics The slides included herein were taken from the materials accompanying Fundamentals of Logic Design, 6th Edition, by Roth and Kinney, and were used with permission from Cengage Learning.

  2. ECE 301 - Digital Electronics Basic Laws and Theorems 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: 5. X + X' = 1 5D. X • X' = 0

  3. ECE 301 - Digital Electronics Basic Laws and Theorems 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 11. (X + Y')Y = XY 11D. XY' + Y = X + Y

  4. ECE 301 - Digital Electronics Basic Laws and Theorems 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 theorem: 15. XY + YZ + X'Z = XY + X'Z 15D. (X + Y)(Y + Z)(X' + Z) = (X + Y)(X' + Z)

  5. ECE 301 - Digital Electronics Use the simplification theorems to simplify the following Boolean expression: F = ABC' + AB'C' + A'BC' Simplification Theorems: Example #1 Simplification Theorems (9 – 11): X.Y + X.Y' = X (X+Y).(X+Y') = X X + X.Y = X X.(X+Y) = X (X+Y').Y = X.Y X.Y' + Y = X+Y

  6. ECE 301 - Digital Electronics Use the simplification theorems to simplify the following Boolean expression: F = (A'+B'+C').(A+B'+C').(B'+C) Simplification Theorems: Example #2 Simplification Theorems (9 – 11): X.Y + X.Y' = X (X+Y).(X+Y') = X X + X.Y = X X.(X+Y) = X (X+Y').Y = X.Y X.Y' + Y = X+Y

  7. ECE 301 - Digital Electronics Use the simplification theorems to simplify the following Boolean expression: F = AB'CD'E + ACD + ACF'GH' +ABCD'E +ACDE' + E'H' Simplification Theorems: Example #3 Simplification Theorems (9 – 11): X.Y + X.Y' = X (X+Y).(X+Y') = X X + X.Y = X X.(X+Y) = X (X+Y').Y = X.Y X.Y' + Y = X+Y (See Programmed Exercise 3.4 on page 75)

  8. ECE 301 - Digital Electronics Use the consensus theorem to simplify the following Boolean expression: F = ABC + BCD + A'CD + B'C'D' Consensus Theorem: Example #1 Consensus Theorem: (15) X.Y + Y.Z + X'.Z = X.Y + X'.Z (15D) (X+Y).(Y+Z).(X'+Z) = (X+Y).(X'+Z)

  9. ECE 301 - Digital Electronics Use the consensus theorem to simplify the following Boolean expression: F = (A+C+D')(A+B'+D)(B+C+D)(A+B'+C) Consensus Theorem: Example #2 Consensus Theorem: (15) X.Y + Y.Z + X'.Z = X.Y + X'.Z (15D) (X+Y).(Y+Z).(X'+Z) = (X+Y).(X'+Z)

  10. ECE 301 - Digital Electronics Use the consensus theorem to simplify the following Boolean expression: F = AC' + AB'D + A'B'C + A'CD' + B'C'D' Consensus Theorem: Example #3 Consensus Theorem: (15) X.Y + Y.Z + X'.Z = X.Y + X'.Z (15D) (X+Y).(Y+Z).(X'+Z) = (X+Y).(X'+Z) (See Programmed Exercise 3.5 on page 77)

  11. ECE 301 - Digital Electronics Find the complement of the following Boolean expression using DeMorgan's law: F = (A + BC').((A'C)' + (D' + E)) DeMorgan's Law: Example DeMorgan's Law: (12) (X + Y + Z + … )' = X'.Y'.Z'... (12D) (X.Y.Z… )' = X' +Y' + Z' …

  12. ECE 301 - Digital Electronics Simplifying Boolean Expressions • Boolean algebra can be used in several ways to simplify a Boolean expression: • Combine terms • Eliminate redundant or consensus terms • Eliminate redundant literals • Add redundant terms to be combined with or allow the elimination of other terms

  13. ECE 301 - Digital Electronics Importance of Boolean Algebra • Boolean algebra is used to simplify Boolean expressions. • Simpler expressions leads to simpler logic circuits. • Reduces cost • Reduces area requirements • Reduces power consumption • The objective of the digital circuit designer is to design and realize optimal digital circuits. • Thus, Boolean algebra is an important tool to the digital circuit designer.

  14. ECE 301 - Digital Electronics Problem with Boolean Algebra • In general, there is no easy way to determine when a Boolean expression has been simplified to a minimum number of terms or a minimum number of literals. • Karnaugh Maps provide a better mechanism for the simplification of Boolean expressions.

  15. ECE 301 - Digital Electronics Circuit Design: Example For the following Boolean expression: F(A,B,C) = A.B.C + A'.B.C + A.B'.C + A.B.C' 1. Draw the circuit diagram 2. Simplify using Boolean algebra 3. Draw the simplified circuit diagram

  16. ECE 301 - Digital Electronics Standard Forms of Boolean Expressions

  17. ECE 301 - Digital Electronics There are two standard forms in which all Boolean expressions can be written: 1. Sum of Products (SOP) 2. Product of Sums (POS) Standard Forms

  18. ECE 301 - Digital Electronics Sum of Products (SOP) • Product Term • Logical product = AND operation • A product term is the ANDing of literals • Examples: A.B, A'.B.C, A.C', B.C'.D', A.B.C.D • “Sum of” • Logical sum = OR operation • The sum of products is the ORing of product terms.

  19. ECE 301 - Digital Electronics Sum of Products (SOP) • The distributive laws are used to multiply out a general Boolean expression to obtain the sum of products (SOP) form. • The distributive laws are also used to convert a Boolean expression in POS form to one in SOP form. • A SOP expression is realized using a set of AND gates (one for each product term) driving a single OR gate (for the sum).

  20. ECE 301 - Digital Electronics Product of Sums (POS) • Sum Term • Logical sum = OR operation • A sum term is the ORing of literals • Examples: A+B, A'+B+C, A+C', B+C'+D' • “Product of” • Logical product = AND operation • The product of sums is the ANDing of sum terms.

  21. ECE 301 - Digital Electronics Product of Sums (POS) • The distributive laws are used to factor a general Boolean expression to obtain the product of sums (POS) form. • The distributive laws are also used to convert a Boolean expression in SOP form to one in POS form. • A POS expression is realized using a set of OR gates (one for each sum term) driving a single AND gate (for the product).

  22. ECE 301 - Digital Electronics For each of the following Boolean expressions, identify whether it is in SOP or POS form: 1. F(A,B,C) = (A+B).(A'+B'+C').(B+C') 2. F(A,B,C) = A.B.C + B'.C' + A.C' + A'.B.C' 3. F(A,B,C) = A + B.C + B'.C' + A'.B'.C 4. F(A,B,C) = (A'+B'+C).(B+C').(A+C').(B') 5. F(A,B,C) = A.B.C + A'.(B+C) + (A+C').B 6. F(A,B,C) = A + B + C SOP and POS: Examples

  23. ECE 301 - Digital Electronics Questions?

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