Boolean Algebra and Digital Logic Design: Concepts and Theorems - Prof. Nur A. Touba, Study notes of Electrical and Electronics Engineering

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EE 316 - Digital Logic Design

Lecture 2

Nur A. Touba

University of Texas at Austin

BOOLEAN ALGEBRA

 Switching Devices

 Two-States

 Special Case of Boolean Algebra Used

• Two-valued “switching” algebra

 George Boole

 Developed Boolean Algebra in 1847

 Claude Shannon

 Applied to switching circuits in 1939

BOOLEAN EXPRESSIONS

 Simplest

 Single Constant: 0, 1

 Single Variable: X, Y’

 More Complicated Expressions

 AB’ + C

 [A(C+D)]’ + BE

 No Parentheses

 Order is Complementation, AND, OR

EVALUATING EXPRESSION

 Substitute value of 0 or 1 for each variable

 [(A+B)D]’ + C’E

TRUTH TABLE

 Truth Table (Table of Combinations)

 Specifies value of Boolean expression

• (^) for every possible combination of values of variables

 Expression with n variables

• (^) 2 n^ rows in truth table

EXAMPLE

LAWS

 Commutative Laws

 XY = YX X+Y = Y+X

 Associative Laws

 (XY)Z = X(YZ) = XYZ

 (X+Y) + Z = X + (Y + Z) = X + Y + Z

 Distributive Law

 X(Y + Z) = XY + XZ

 X + YZ = (X + Y)(X + Z)

PROOF OF ASSOCIATIVE LAW

SIMPLIFICATION THEOREMS

 Useful Theorems for Simplifying Boolean Expressions

 XY + XY’ = X (X+Y)(X+Y’) = X

 X + XY = X X(X + Y) = X

 (X + Y’)Y = XY XY’ + Y = X + Y

SIMPLIFICATION EXAMPLES

 Simplify Z = A’BC + A’

 Simplify Z = [A + B’C + D + EF] [A + B’C + (D + EF)’]

 Simplify Z = (AB + C)(B’D + C’E’) + (AB + C)’

MULTIPYING OUT AND FACTORING

 Product of Sums

 All Sums are sums of single variables

DeMORGAN’s LAWS

 DeMorgan’s Laws

 (X + Y)’ = X’Y’

• (^) Complement of sum is product of complements

 (XY)’ = X’ + Y’

• (^) Complement of product is sum of complements