Logical equivalence Equivalence proof Multiplexers January 22 2020

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Logical equivalence Equivalence proof Multiplexers January 22, 2020 Patrice Belleville / Geoffrey Tien 1

Logical equivalence Equivalence proof Multiplexers January 22, 2020 Patrice Belleville / Geoffrey Tien 1

Announcements • Pre-class quiz #4, due Sunday, Jan. 26, 19: 00 – Epp 4

Announcements • Pre-class quiz #4, due Sunday, Jan. 26, 19: 00 – Epp 4 e/5 e: Chapter 2. 3 • Pre-class quiz #5, due Sunday, Feb. 02, 19: 00 – Epp 4 e/5 e: Chapter 3. 1, 3. 3 • HW 1 due tomorrow, 19: 00 – Start the submission process early – there are several steps which may take a few minutes January 22, 2020 Patrice Belleville / Geoffrey Tien 2

Summary of equivalence rules Name Rule Name Identity laws Universal bounds laws Idempotent laws

Summary of equivalence rules Name Rule Name Identity laws Universal bounds laws Idempotent laws Commutative laws Associative laws Distributive laws Absorption laws Negation laws Doublenegative law De. Morgan's laws Rule Contrapositive law January 22, 2020 Patrice Belleville / Geoffrey Tien 3

Logical equivalence proofs • How do we write a logical equivalence proof? – State

Logical equivalence proofs • How do we write a logical equivalence proof? – State theorem we want to prove – Indicate the beginning of the proof by "Proof: " – Start with one side and work towards the other • one step at a time • without forgetting to justify each step • usually we will simplify the more complicated proposition instead of trying to complicate the simpler one – Indicate the end of the proof by "QED" or January 22, 2020 Patrice Belleville / Geoffrey Tien 4

Logical equivalence proofs • January 22, 2020 Patrice Belleville / Geoffrey Tien 5

Logical equivalence proofs • January 22, 2020 Patrice Belleville / Geoffrey Tien 5

Logic equivalence proofs • January 22, 2020 Patrice Belleville / Geoffrey Tien 6

Logic equivalence proofs • January 22, 2020 Patrice Belleville / Geoffrey Tien 6

Modeling circuits with propositional logic • Propositional logic is not a perfect model of

Modeling circuits with propositional logic • Propositional logic is not a perfect model of how logic gates work • To understand why, we will look at a multiplexer (MUX) • A multiplexer selects one of several possible input signals, and "steers" it to a single output – Analogy: An HDMI switchbox for your TV January 22, 2020 Patrice Belleville / Geoffrey Tien 7

Truth table January 22, 2020 a b sel m F F F T T

Truth table January 22, 2020 a b sel m F F F T T F F T T T Patrice Belleville / Geoffrey Tien 8

Implementation • • Seems reasonable, but this might not work as expected January 22,

Implementation • • Seems reasonable, but this might not work as expected January 22, 2020 Patrice Belleville / Geoffrey Tien 9

Gate delay • January 22, 2020 Patrice Belleville / Geoffrey Tien 10

Gate delay • January 22, 2020 Patrice Belleville / Geoffrey Tien 10

Gate delay T a) b) c) d) e) 10 ns F T 20 ns

Gate delay T a) b) c) d) e) 10 ns F T 20 ns 30 ns 40 ns It may never happen January 22, 2020 F T T T Patrice Belleville / Geoffrey Tien 11

Gate delay • T T F F T T F January 22, 2020 Patrice

Gate delay • T T F F T T F January 22, 2020 Patrice Belleville / Geoffrey Tien 12

Gate delay • T T F F January 22, 2020 Patrice Belleville / Geoffrey

Gate delay • T T F F January 22, 2020 Patrice Belleville / Geoffrey Tien 13

Gate delay • T T F F F The output became F January 22,

Gate delay • T T F F F The output became F January 22, 2020 Patrice Belleville / Geoffrey Tien 14

Gate delay • T T T F F The output became T again January

Gate delay • T T T F F The output became T again January 22, 2020 Patrice Belleville / Geoffrey Tien 15

Instability • January 22, 2020 Patrice Belleville / Geoffrey Tien 16

Instability • January 22, 2020 Patrice Belleville / Geoffrey Tien 16

Instability • January 22, 2020 Patrice Belleville / Geoffrey Tien 17

Instability • January 22, 2020 Patrice Belleville / Geoffrey Tien 17

Multiplexers • Here's a multiplexer that avoids the instability January 22, 2020 Patrice Belleville

Multiplexers • Here's a multiplexer that avoids the instability January 22, 2020 Patrice Belleville / Geoffrey Tien 18

More exercises • January 22, 2020 Patrice Belleville / Geoffrey Tien 19

More exercises • January 22, 2020 Patrice Belleville / Geoffrey Tien 19