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ECE 331 – Digital System Design
Boolean Algebra and Standard Forms of Boolean Expressions (Lecture #4) 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.

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 Spring 2011 ECE Digital System Design

Basic Laws and Theorems
Commutative laws: 6. X + Y = Y + X D. 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 D. (X + Y)(X + Y') = X 10. X + XY = X D. X(X + Y) = X 11. (X + Y')Y = XY D. XY' + Y = X + Y Spring 2011 ECE Digital System Design

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 D. (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) Spring 2011 ECE Digital System Design

ECE 331 - Digital System Design
Duality (13) The dual of a Boolean expression can be written by Replacing AND with OR, and OR with AND Replacing 0 with 1, and 1 with 0 Leaving literals unchanged See the Boolean laws and theorems, previously discussed, for examples of Boolean expressions and their duals. Spring 2011 ECE Digital System Design

Distributive Law: Example #1
Distributive law (8): X.(Y + Z) = X.Y + X.Z Use the distributive law to multiply out the following Boolean expression: F = (A+B).(C+D).(E+F) Spring 2011 ECE Digital System Design

Distributive Law: Example #2
Distributive law (8D): X + Y.Z = (X+Y).(X+Z) Use the distributive law to factor the following Boolean expression: F = A.B + C.D Spring 2011 ECE Digital System Design

Simplification Theorems: Example #1
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 Use the simplification theorems to simplify the following Boolean expression: F = ABC' + AB'C' + A'BC' Spring 2011 ECE Digital System Design

Simplification Theorems: Example #2
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 Use the simplification theorems to simplify the following Boolean expression: F = (A'+B'+C').(A+B'+C').(B'+C) Spring 2011 ECE Digital System Design

Simplification Theorems: Example #3
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 Use the simplification theorems to simplify the following Boolean expression: F = AB'CD'E + ACD + ACF'GH' +ABCD'E +ACDE' + E'H' (See Programmed Exercise 3.4 on page 75) Spring 2011 ECE Digital System Design

Consensus Theorem: Example #1
(15) X.Y + Y.Z + X'.Z = X.Y + X'.Z (15D) (X+Y).(Y+Z).(X'+Z) = (X+Y).(X'+Z) Use the consensus theorem to simplify the following Boolean expression: F = ABC + BCD + A'CD + B'C'D' Spring 2011 ECE Digital System Design

Consensus Theorem: Example #2
(15) X.Y + Y.Z + X'.Z = X.Y + X'.Z (15D) (X+Y).(Y+Z).(X'+Z) = (X+Y).(X'+Z) Use the consensus theorem to simplify the following Boolean expression: F = (A+C+D')(A+B'+D)(B+C+D)(A+B'+C) Spring 2011 ECE Digital System Design

Consensus Theorem: Example #3
(15) X.Y + Y.Z + X'.Z = X.Y + X'.Z (15D) (X+Y).(Y+Z).(X'+Z) = (X+Y).(X'+Z) Use the consensus theorem to simplify the following Boolean expression: F = AC' + AB'D + A'B'C + A'CD' + B'C'D' (See Programmed Exercise 3.5 on page 77) Spring 2011 ECE Digital System Design

ECE 331 - Digital System Design
DeMorgan's Law DeMorgan's Law: (12) (X + Y + Z + … )' = X'.Y'.Z'... (12D) (X.Y.Z… )' = X' +Y' + Z' … Prove (using a truth table): (X+Y)' = X'.Y' X Y X + Y (X + Y)' X' Y' X'.Y' 1 Spring 2011 ECE Digital System Design

ECE 331 - Digital System Design
DeMorgan's Law Graphical representation of DeMorgan's Law x y (X.Y)' X' + Y' (X+Y)' X'.Y' Spring 2011 ECE Digital System Design

DeMorgan's Law: Example
(12) (X + Y + Z + … )' = X'.Y'.Z'... (12D) (X.Y.Z… )' = X' +Y' + Z' … Find the complement of the following Boolean expression using DeMorgan's law: F = (A + (BC)').((AD)' + C.(B' + D)) Spring 2011 ECE Digital System Design

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 Spring 2011 ECE Digital System Design

Equivalency of Boolean Expressions
Two Boolean expressions are equivalent iff both expressions evaluate to the same value for all combinations of the variables in the expressions. The equivalency can be proven using A Truth table Boolean algebra theorems to manipulate one expression until it is identical to the other. Boolean algebra theorems to reduce both expressions independently to the same expression. Spring 2011 ECE Digital System Design

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. Spring 2011 ECE Digital System Design

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. Spring 2011 ECE Digital System Design

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 Spring 2011 ECE Digital System Design

Standard Forms of Boolean Expressions
Spring 2011 ECE Digital System Design

ECE 331 - Digital System Design
Standard Forms There are two standard forms in which all Boolean expressions can be written: 1. Sum of Products (SOP) 2. Product of Sums (POS) Spring 2011 ECE Digital System Design

ECE 331 - Digital System Design
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. Spring 2011 ECE Digital System Design

ECE 331 - Digital System Design
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). Spring 2011 ECE Digital System Design

ECE 331 - Digital System Design
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. Spring 2011 ECE Digital System Design

ECE 331 - Digital System Design
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). Spring 2011 ECE Digital System Design

ECE 331 - Digital System Design
SOP and POS: Examples 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 Spring 2011 ECE Digital System Design

ECE 331 - Digital System Design
Questions? Spring 2011 ECE Digital System Design

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