6.042J/18.062J 1. Good and Bad Proofs

1 February 3, 2010 Albert R. Meyer, 2010 lec 1W.1 Mathematics for Computer Science 6.042J/18.062J http://courses.csail.mit.edu/6.042 WELCOME! Prof. Albert R Meyer February 3, 2010 Albert R. Meyer, 2010 lec 1W.2 Quick Summary 1. Fundamental Concepts of Discrete Mathematics ( sets, relations, proof methods,… ) 2. Discrete Mathematical Structures ( graphs, trees, counting…) 3. Discrete Probability Theory February 3, 2010 Albert R. Meyer, 2010 lec 1W.3 Vocabulary Quickie: What does “discrete” mean? ( ≠ “discreet”) February 3, 2010 Albert R. Meyer, 2010 lec 1W.4 Online Tutor Registration • TP.1: Registration asap • not later than Saturday, 9AM for table assignment February 3, 2010 Albert R. Meyer, 2010 lec 1W.5 Reading Assignment • Courseinfo on web page asap • Notes Chapters 1 & 2 asap • Ch. 3, 4.1—4.7 next week • Reading Comments --using online system NB February 3, 2010 Albert R. Meyer, 2010 lec 1W.6 Course Web site http://courses.csail.mit.edu/6.042 • announcements • class schedule • notes, slides,… • course organization • grading info 2 February 3, 2010 Albert R. Meyer, 2010 lec 1W.7 Lecture & Team Problems Three 1.5 hour class sessions: • 1/2 hour overview lecture, • then team problem-solving. Team participation counts 20% of final grade Teams assigned by Monday February 3, 2010 Albert R. Meyer, 2010 lec 1W.9 Active Lectures Say “hello” to your neighbors –you’ll be working with them February 3, 2010 Albert R. Meyer, 2010 lec 1W.10 Active Lectures Quickie question: Where was your neighbor born? February 3, 2010 Albert R. Meyer, 2010 lec 1W.11 Getting started: Pythagorean theorem Familiar? Obvious? c b a Yes! No! a2 +b2 = c2 February 3, 2010 Albert R. Meyer, 2010 lec 1W.12 A Cool Proof c b a Rearrange into: (i) a c × c square, and then (ii) an a×a & a b×b square February 3, 2010 Albert R. Meyer, 2010 lec 1W.13 A Cool Proof c c c a b c b-a 3 February 3, 2010 Albert R. Meyer, 2010 lec 1W.15 A Cool Proof b a a a b-a February 3, 2010 Albert R. Meyer, 2010 lec 1W.16 A False Proof: Getting Rich By Diagram 11 10 11 1 1 1 1 1 1 1 1 10 February 3, 2010 Albert R. Meyer, 2010 lec 1W.17 1 1 Profit! 10 10 11 1 1 1 1 11 A False Proof: Getting Rich By Diagram February 3, 2010 Albert R. Meyer, 2010 lec 1W.18 The bug: are not right triangles! So the top and bottom line of the “rectangle” is not straight! Getting Rich 10 1 1 1 1 1 February 3, 2010 Albert R. Meyer, 2010 lec 1W.23 1 = -1 ? pictures are not the only source of false proofs Moral: 1. Calculation is a risky substitute for understanding. 2. Be sure you know the rules. February 3, 2010 Albert R. Meyer, 2010 lec 1W.25 ½ = -½ (multiply by ½) 2 = 1 (add ) Consequences of 1= -1 “Since I and the Pope are clearly 2, we conclude that I and the Pope are 1. That is, I am the Pope.” --Bertrand Russell 4 February 3, 2010 Albert R. Meyer, 2010 lec 1W.26 Consequences of 1= -1 Bertrand Russell (1872 -1970) (Picture source: http://www.users.drew.edu/~jlenz/brs.html) February 3, 2010 Albert R. Meyer, 2010 lec 1W.27 Team Problems Problems 1‒3 Image removed due to copyright restrictions.MIT OpenCourseWare http://ocw.mit.edu 6.042J /18.062J Mathematics for Computer Science Spring 2010 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.