Logical equivalence Equivalence proof Multiplexers January 22 2020
- Slides: 19
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 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 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 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
Logic equivalence proofs • January 22, 2020 Patrice Belleville / Geoffrey Tien 6
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 F F T T T Patrice Belleville / Geoffrey Tien 8
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 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 Belleville / Geoffrey Tien 12
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, 2020 Patrice Belleville / Geoffrey Tien 14
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 17
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
- Logical equivalence in discrete mathematics
- Logical form
- What is decoder in computer architecture
- Decoders and multiplexers
- Multiplexers and decoders
- Multiplexers and decoders
- Indirect proof assumption
- Direct proof and indirect proof
- Direct proof and indirect proof
- Algebraic proof
- Direct proof and indirect proof
- Algebraic proof definition
- Logical equivalence cheat sheet
- Red herring fallacy
- Disjunctive normal form
- Implication law
- Applications of propositional logic in discrete mathematics
- Negation law
- Algebra of propositions
- Logical equivalence