Lecture03

Lecture03 - Lecture 3 Spring 2010 1 ECE 2300 Introduction...

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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) 0•0=0 (A3’) 1+1=1 (A4) 1•1=1 (A4’) 0+0=0 (A5) 0•1=1•0=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 W•X+Y•Z mean? – W•((X+Y)•Z) ? – (W•(X+Y))•Z ? – (W•X)+(Y•Z) ? • As in normal arithmetic, “multiplication” (AND) has precedence over “addition” (OR) – (W•X) + (Y•Z) – Can use parentheses to avoid ambiguity Lecture 3:10 Single-Variable Theorems • Identities: (T1) X+0=X (T1’) X•1=X • Null Elements: (T2) X+1=1 (T2’) X•0=0...
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This note was uploaded on 04/10/2010 for the course ECE 2300 taught by Professor Long during the Spring '08 term at Cornell.

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Lecture03 - Lecture 3 Spring 2010 1 ECE 2300 Introduction...

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