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Unformatted text preview: Lecture 3: Spring 2010 1 ECE 2300 Introduction to Digital Logic Design Switching Algebra Lecture 3: 2 Announcements HW 1 due this Wednesday at noon In the HW drop box Lab 1 is on Blackboard Prelab is due this Friday at noon Electronic submission (details in lab write-up) Instructor and TA office hours are on Blackboard Lecture 3: 3 Switching Algebra Mathematical tool for analyzing logic circuits Based on Boolean algebra (George Boole, 1854) Two-valued algebraic system Used to formulate true or false postulations Switching algebra (Claude Shannon, 1938) Adopted Boolean algebra for digital circuits Lecture 3: 4 Definitions Literal : a variable or its complement X, X, val Expression : literals combined by AND, OR, complementation (not), parentheses P Q R A + B C ((V Z) + A B C + Q5) RESET Equation : Variable = expression P = ((V Z) + A B C + Q5) RESET Lecture 3: 5 Complement: X AND: X Y OR: X + Y Boolean operators Lecture 3: 6 Axioms of Switching Algebra Axioms come in pairs Interchange 1 and 0, AND and OR First axiom: Digital abstraction A variable X can only take on 2 values (A1) X = 0 if X ! 1 (A1) X = 1 if X ! Lecture 3: 7 X Y = X Other complement symbols: ~X, /X, X Axioms of Switching Algebra Complement (A2) If X = 0, then X = 1 (A2) If X = 1, then X = 0 Lecture 3: 8 Formal definitions of AND and OR (A3) 00=0 (A3) 1+1=1 (A4) 11=1 (A4) 0+0=0 (A5) 01=10=0 (A5) 1+0=0+1=1 A1-A5 completely define switching algebra Everything else can be derived from these axioms Axioms of Switching Algebra Lecture 3: 9 Operator Precedence What does WX+YZ mean? W((X+Y)Z) ? (W(X+Y))Z ? (WX)+(YZ) ? As in normal arithmetic, multiplication (AND) has precedence over addition (OR) (WX) + (YZ) Can use parentheses to avoid ambiguity Lecture 3:10 Single-Variable Theorems Identities: (T1) X+0=X (T1) X1=X Null Elements: (T2) X+1=1 (T2) X0=0...
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- Spring '08