lect3-Boolean algebra & Combinational logic

# lect3-Boolean algebra & Combinational logic - CSE140:...

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1 1 CSE140: Components and Design Techniques for Digital Systems Boolean algebra & Combinational logic Tajana Simunic Rosing 2 Overview • Reminder: HW#1 is due today! • What we covered thus far: – Number representations – Switches –MOS t r a n s i s t o r s – Logic gates • What is next: – Boolean algebra – Combinatorial logic: • Representations • Minimization • Implementations

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2 3 Review: CMOS Example • Remember the rules: – NMOS connects to GND, PMOS to power supply Vdd – Duality of NMOS and PMOS – Rp ~ 3 Rn => PMOS in series is much slower than NMOS in series • Implement Z using CMOS: Z = (A + BC)’ 4 Another CMOS Example • Implement F using CMOS: F=A*(B+C)
3 5 Boolean Algebra and its Relation to Digital Circuits • An algebraic structure consists of – a set of elements B – binary operations { + , • } (OR and AND) – and a unary operation { ’ } (NOT) – such that the following axioms hold: 1. the set B contains at least two elements: a, b 2. closure: a + b is in B a • b is in B 3. commutativity: a + b = b + a a • b = b • a 4. associativity: a + (b + c) = (a + b) + c a • (b • c) = (a • b) • c 5. identity: a + 0 = aa • 1 = a 6. distributivity: a + (b • c) = (a + b) • (a + c) a • (b + c) = (a • b) + (a • c) 7. complementarity: a + a’ = 1 a • a’ = 0 6 Axioms and theorems of Boolean algebra identity 1. X + 0 = X 1D. X • 1 = X null 2. X + 1 = 1 2D. X • 0 = 0 idempotency: 3. X + X = X 3D. X • X = X involution: 4. (X’)’ = X complementarity: 5. X + X’ = 1 5D. X • X’ = 0 commutativity: 6. X + Y = Y + X 6D. X • Y = Y • X associativity: 7. (X + Y) + Z = X + (Y + Z) 7D. (X • Y) • Z = X • (Y • Z)

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4 7 Axioms and theorems of Boolean algebra (cont’d) distributivity: 8. X • (Y + Z) = (X • Y) + (X • Z) 8D. X + (Y • Z) = (X + Y) • (X + Z) uniting: 9. X • Y + X • Y’ = X 9D. (X + Y) • (X + Y’) = X absorption: 10. X + X • Y = X 10D. X • (X + Y) = X 11. (X + Y’) • Y = X • Y 11D. (X • Y’) + Y = X + Y factoring: 12. (X + Y) • (X’ + Z) = 12D. X • Y + X’ • Z = X • Z + X’ • Y (X + Z) • (X’ + Y) concensus: 13. (X • Y) + (Y • Z) + (X’ • Z) = 13D. (X + Y) • (Y + Z) • (X’ + Z) = X • Y + X’ • Z (X + Y) • (X’ + Z) de Morgan’s: 14. (X + Y + . ..)’ = X’ • Y’ • . .. 14D. (X • Y • . ..)’ = X’ + Y’ +…
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## This note was uploaded on 02/14/2008 for the course CSE 140 taught by Professor Rosing during the Fall '06 term at UCSD.

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lect3-Boolean algebra & Combinational logic - CSE140:...

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