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Boolean Algebra and Expressions in Digital System Design

Learn how Boolean algebra can be used to manipulate and simplify Boolean expressions, resulting in more cost-effective circuit designs.

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Boolean Algebra and Expressions in Digital System Design

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  1. Boolean Algebra (Lecture #3) ECE 331 – Digital System Design 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 331 - Digital System Design Boolean Algebra • George Boole developed an algebraic description for processes involving logical thought and reasoning. • Became known as Boolean Algebra • Claude Shannon later demonstrated that Boolean Algebra could be used to describe switching circuits. • Switching circuits are circuits built from devices that switch between two states (e.g. 0 and 1). • Switching Algebra is a special case of Boolean Algebra in which all variables take on just two distinct values • Boolean Algebra is a powerful tool for analyzing and designing logic circuits.

  3. ECE 331 - Digital System Design Boolean Algebra and Expressions • A logic function can be represented by a • Boolean expression (i.e. equation) • Truth table • Circuit diagram • A Boolean expression is composed of logic operations and literals • where the logic operations are represented by logic gates in the circuit diagram • and the literals are represented by the inputs to the logic gates.

  4. ECE 331 - Digital System Design Boolean algebra can be used to manipulate or simplify Boolean expressions. Why is this useful? Boolean Algebra and Expressions

  5. ECE 331 - Digital System Design Boolean Algebra and Expressions • Manipulating a Boolean expression (using Boolean algebra) results in an alternate expression that is functionally equivalent to the original. • The circuit diagram corresponding to the new expression may be • Easier to build (due to the logic gates required) than the circuit diagram corresponding to the original expression. • More cost effective than the circuit diagram corresponding to the original expression.

  6. ECE 331 - Digital System Design Boolean Algebra and Expressions • Simplifying a Boolean expression (using Boolean algebra) results in an expression with fewer logic operations and/or fewer literals than the original. • The two expressions are functionally equivalent. • The circuit diagram corresponding to the new expression will be • More cost effective than the circuit diagram corresponding to the original expression.

  7. ECE 331 - Digital System Design Basic Laws and Theorems

  8. ECE 331 - Digital System Design 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

  9. ECE 331 - Digital System Design 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

  10. ECE 331 - 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... 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)

  11. ECE 331 - Digital System Design Distributive Law Using a truth table, it is easy to show that the distributive law is valid: In addition to the ordinary distributive law, a second distributive law is valid for Boolean algebra, but not for ordinary algebra: Proof of the second distributive law follows:

  12. ECE 331 - Digital System Design Multiplying Out • The two distributive laws are used to multiply out an expression to obtain a Sum-of-Products (SOP) form. • An expression is said to be in SOP form when all products are the products of single variables. • A SOP expression is realized using one or more AND gates feeding a single OR gate.

  13. ECE 331 - Digital System Design Examples of Boolean expression in SOP: F1(A,B,C) = A'BC + ABC' + A'B'C' + AC F2(A,B,C) = ABC + A'B'C' + B'C Examples of Boolean expressions not in SOP: F3(A,B,C) = A'.(B+C) + A.B' + A.B.C Multiplying Out

  14. ECE 331 - Digital System Design Factoring • The two distributive laws can be used to factor an expression to obtain a Product-of-Sums (POS) form. • An expression is said to be in POS form when all sums are the sums of single variables. • A POS expression is realized using one or more OR gates feeding a single AND gate.

  15. ECE 331 - Digital System Design Examples of Boolean expression in POS: F1(A,B,C) = (A'+B+C).(A+B'+C').(A+B+C).(A'+B') F2(A,B,C) = (A+B+C').(A'+B'+C).(B'+C') Examples of Boolean expressions not in POS: F3(A,B,C) = (A'.B+C).(A'+B').(A+B'+C') Factoring

  16. ECE 331 - Digital System Design Simplification Theorems The following theorems are useful in simplifying Boolean expressions:

  17. ECE 331 - Digital System Design Simplifying a Logic Circuit F = A(A’ + B) By Theorem (2-14), (X + Y’)Y = XY, the expression F simplifies to AB. Which circuit is cheaper?

  18. ECE 331 - Digital System Design Example: Using Boolean Algebra, simplify the following Boolean Expression: F = A'.B.C + A' Boolean Algebra

  19. ECE 331 - Digital System Design Example: Using Boolean Algebra, simplify the following Boolean Expression: F = (A + B'C + D + EF)(A + B'C + (D + EF)') Boolean Algebra

  20. ECE 331 - Digital System Design Exercise: Using Boolean Algebra, simplify the following Boolean Expression: F = (AB + C)(B'D + C'E') + (AB + C)' Boolean Algebra

  21. ECE 331 - Digital System Design Exercise: Using Boolean Algebra, simplify the following Boolean expression. F(A,B,C) = A'.B.C + A.B'.C + A.B.C Boolean Algebra

  22. ECE 331 - Digital System Design DeMorgan's Laws • Can be stated as follows: • The complement of the product (AND) is the sum (OR) of the complements. • (X.Y)' = X' + Y' • The complement of the sum (OR) is the product (AND) of the complements. • (X + Y)' = X' . Y' • Easily generalized to n variables. • Can be proven using a Truth table

  23. ECE 331 - Digital System Design Proving DeMorgan's Law (X . Y)' = X' + Y'

  24. ECE 331 - Digital System Design DeMorgan's Laws x 1 x x 1 1 x x 2 2 x 2 x x x x = + (a) 1 2 1 2 x 1 x x 1 1 x x 2 2 x 2 x x x x + = (b) 1 2 1 2

  25. ECE 331 - Digital System Design Example: Using DeMorgan's Laws, find the complement of the following Boolean Expression: F = (A' + B).C' DeMorgan's Laws

  26. ECE 331 - Digital System Design Exercise: Using DeMorgan's Laws, find the complement of the following Boolean Expression: F = (AB' + C).D' + E' DeMorgan's Laws

  27. ECE 331 - Digital System Design (XYZ ...)D = X + Y + Z + ... (X + Y + Z + ...)D = XYZ ... Duality Given an expression, the dual is formed by replacing AND with OR, OR with AND, 0 with 1, and 1 with 0. Variables and complements are left unchanged. The dual of AND is OR and the dual of OR is AND:

  28. ECE 331 - Digital System Design Duality The dual of an expression may be found by complementing the entire expression and then complementing each of the individual variables. For example: (AB' + C)' = (AB')'.C' = (A' + B).C' Therfore, (AB' + C)D = (A + B').C complement expression complement individual variables

  29. ECE 331 - Digital System Design Exercise: For the following Boolean expression F(A,B,C) = ABC + A'BC + AB'C + ABC' 1. Using Boolean algebra, simplify the Boolean expression. 2. Derive the Truth table for the simplified expression. 3. Confirm that the Truth table is also valid for the original expression. 4. Draw the circuit diagram for the “cheaper” expression. Boolean Algebra

  30. ECE 331 - Digital System Design Questions?

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