Class 1: What makes it go?

cs2102: Discrete Mathematics
University of Virginia, Fall 2017

See course site for notes:
https://uvacs2102.github.io

Transcript

1. cs2102: Discrete Mathematics Photo: Casey Kilmartin
2. Photo: Casey Kilmartin
3. Class 1: What makes computers go? I’ll have (brief) office hours after class today (until 4pm) and tomorrow (Wednesday), 3-4pm [Rice 507] TA’s office hours will start next week (schedule to be posted on course site) cs2102: Discrete Mathematics
4. What makes it go?
5. For example, there was a book that started out with four pictures: first there was a windup toy; then there was an automobile; then there was a boy riding a bicycle; then there was something else. And underneath each picture it said, “What makes it go?” From Richard Feynman, Surely You’re Joking, Mr. Feynman!
6. I thought, “I know what it is: They’re going to talk about mechanics, how the springs work inside the toy; …” It was the kind of thing my father would have talked about: “What makes it go? Everything goes because the sun is shining.” … “No, the toy goes because the spring is wound up,” I would say. “How did the spring get wound up?" … “I wound it up." “And how did you get moving?" “From eating." “And food grows only because the sun is shining. So it’s because the sun is shining that all these things are moving.”… From Richard Feynman, Surely You’re Joking, Mr. Feynman!
7. What makes it go?
8. Sun shines Plankton thrives Fossilizes
9. Electricity Transistors, Memory
10. What makes it go?
11. Transistors, Memory Programs Mathematics Algorithms
12. Focus of cs2102 This course is about the underlying mathematics that makes computers work. cs2102 cs3102 cs4102 Theory of Computation: What Can and Cannot Be Computed Algorithms: Designing and Analyzing Information Processes
13. Developing Mathematical Skills Precise Tools for Thinking: Definitions Proofs Before Friday (questions on Survey): “Habits of highly mathematical people”
14. Developing Mathematical Skills Precise Tools for Thinking: Definitions Proofs Mathematical objects used in Computing: Numbers, Sets, Graphs
15. Join the course slack group: https://csmath.slack.com/signup
16. Join the course slack group: csmath.slack.com/signup Post any questions/comments you have so far in #general Answer in #inclass: why does computer science mostly use discrete math? Any other thoughts or jokes you have in #random
17. Course Logistics Read the syllabus before Thursday’s class and post your questions in slack. We won’t talk more about course logistics unless you ask.
18. Two Professors! Mohammad Mahmoody David Evans
19. ~2# Assistant Teachers
20. Assignments and Exams Problem Sets - due most Fridays (6:29pm) Main way to learn this material is to solve problems Can work in groups and discuss together, but must write up your own solutions Designed for learning Grading not meant to be stressful! Exams Two in-class exams (Oct 5, Nov 9), Final (Dec 7)
21. Help Office Hours Me: after class today (-4pm), tomorrow (Weds, 3-4pm) Lots more will be scheduled (see website) Come early – don’t wait until day or two before assignment is due don’t wait until near exam - concepts build on each other This course will require you to learn new and different ways of thinking. This can be overwhelming and it is hard to catch up if you fall behind. Please take advantage of available help!
22. Help On-Line Slack: use this for public questions #general #inclass you can use a pseudonym in slack can use direct messages for private questions Email: use this for private questions This course will require you to learn new and different ways of thinking. This can be overwhelming and it is hard to catch up if you fall behind. Please take advantage of available help!
23. Why Discrete Math? Read the syllabus before Thursday’s class and post your questions in slack. We won’t talk more about course logistics unless you ask.
24. Continuous Discrete
25. Continuous Discrete There is something between any two numbers There is nothing between adjacent numbers Real numbers Integers
26. Represents a 1 Represents a 0 Digital Abstraction 0 V 5 V 2 V 3 V
27. Represents a 1 Represents a 0 Digital Abstraction 0 V 5 V 2 V 3 V This is wasteful - infinitely many values used to represent only two! But useful: tolerate noise and get exact results with confidence.
28. Finite Memory Discrete math is often infinite, but also often done over finite domains.
29. Finite Memory Can continuous math be finite?
30. “Hard” Problems y2 = x3 + 7 What does y2 = x3 + 7 look like in finite field?
31. “Hard” Problems What does y2 = x3 + 7 look like in finite field? 2 = 3 + 7 mod ( = 629)
32. “Hard” Problems Why do we want “hard” problems? 2 = 3 + 7 mod ( = 629)
33. 2 = 3 + 7 mod ( = 2256 − 232 − 977) Bitcoin Market Value Depends on it being hard to find points on this (discrete) curve.
34. All* Discreet Communications depend on Discrete Math * Except for whispering and passing notes. Me mod n YA =aXA mod q Diffie-Hellman Merkle Key Agreement: Discrete log problem is hard RSA Public-Key Encryption: Factoring problem is hard
35. Definitions
36. What makes a good definition?
37. Proposition A proposition is a statement that is either true or false.
38. Proposition A proposition is a statement that is either true or false. The cs2102 TAs are awesome. 2 + 2 = 5. Today is Tuesday. If today is Tuesday and it is during the semester, there is cs2102 class.
39. Axioms An axiom is a proposition that is accepted to be true. Axioms are assumed, not proven.
40. Euclid’s Axioms 1. A straight line may be drawn between any two points. 2. A piece of straight line may be extended indefinitely. 3. A circle may be drawn with any given radius and an arbitrary center. 4. All right angles are equal. 5. If a straight line crossing two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if extended indefinitely, meet on that side on which are the angles less than the two right angles.
41. Mathematical Proof Axioms P Chain of steps using inference rules, starting from set of axioms, resulting in P.
42. Inference Rule: Modus Ponens antecedents conclusion Variables: P, Q
43. Soundness An inference rule is sound, if it can only be used to conclude true things. antecedents conclusion
44. Soundness An inference rule is unsound, if there is a way to make its antecedents all true, and its conclusion false. antecedents conclusion
45. Is it sound? P Q No: It is unsound since setting = true, = false would allow drawing conclusion false.
46. Is it sound? P, P implies Q false No: It is unsound since setting = true, = true would allow drawing conclusion false.
47. Is it sound? P, NOT(P) false Yes! It is sound since there is no choice for the variables that would lead to a false conclusion. There are two cases: = true: NOT() is false, so antecedents are not satisfied and rule cannot be applied = false: is false, so antecedents are not satisfied and rule cannot be applied
48. Charge Before Thursday’s class: read MCS chapter 1, course syllabus; post questions on slack Before Friday (6:29pm): course pledge, registration survey (read “Habits of Highly Mathematical People”) I’ll have (brief) office hours now (until 4pm) and tomorrow (Wednesday), 3-4pm [Rice 507] TA’s office hours will start next week (schedule to be posted on course site)