CMSC 250 - Discrete Structures Summer 2016 Jason Filippou UMCP 05-31-2016 Jason Filippou (UMCP) Discrete Structures 05-31-2016 1 / 38
Outline 1 Overview & Logistics 2 Subject of the course Short history of Discrete Mathematics As a Computer Scientist... As a CS-UMD student... 3 What we’ll (tentatively) cover Jason Filippou (UMCP) Discrete Structures 05-31-2016 2 / 38
Overview & Logistics Overview & Logistics Jason Filippou (UMCP) Discrete Structures 05-31-2016 3 / 38
Overview & Logistics Course overview Webpage: http://cs.umd.edu/class/summer2016/cmsc250/ May 31-July 22 Expected days “lost”: July 4, 2 Midterm days (06-17, 07-08), Final day (07-22). Look at the syllabus for policy on excused absences, academic honesty, etc Please register on Piazza! TAs: Parsa Saadatpanah , Yancy Liao . Office hours have been posted! Jason Filippou (UMCP) Discrete Structures 05-31-2016 4 / 38
Overview & Logistics Course overview Textbooks (recommended): “Discrete Mathematics and Applications”, Susanna Epp, any edition ≥ 2nd. (UMD standard, expensive to buy new). “Discrete Mathematics and Applications”, Thomas Koshy, 1st edition (cheaper, more in line with our flow). Bookstore should have a small number for rentals. Grading (subject to minor changes): 5 homework assignments: 15% (3% each) 5 quizzes: 10% (2% each) 2 in-class midterms: 20 & 25% respectively Final (comprehensive, in-class): 30% Jason Filippou (UMCP) Discrete Structures 05-31-2016 5 / 38
Overview & Logistics Requirements No exceptional CS / mathematical background required for the course. Advanced highschool math material (Calculus, Probability, Set Theory) helpful, but not required. Charlie the Unicorn requirement: All students are required to watch this 20- minute video outlining the epic saga of “Charlie the Unicorn” and submit a half-page essay on their favorite parts of the series . We will be using elements of Charlie’s story in the early parts of the course to explain aspects of “Predicate” Logic. Figure 1: Charlie, pictured here in between Purple and Blue Unicorns, quite distressed. Jason Filippou (UMCP) Discrete Structures 05-31-2016 6 / 38
Overview & Logistics Your Instructor Figure 2: If I do this trip one more time Greek-Canadian States: 2012-today PhD, CompSci Probabilistic Graphical Models, Action Recognition, . . . Expected graduation: ??? Likes: Coffee Dislikes: Everything else Jason Filippou (UMCP) Discrete Structures 05-31-2016 7 / 38
Overview & Logistics My school D.I.T, N K UA (not N T UA) Crappy buildings and infrastructure, great professors. Quite strong in: Databases / Data Mining Theory Logic Jason Filippou (UMCP) Discrete Structures 05-31-2016 8 / 38
Overview & Logistics My hometown Jason Filippou (UMCP) Discrete Structures 05-31-2016 9 / 38
Overview & Logistics My hometown Jason Filippou (UMCP) Discrete Structures 05-31-2016 10 / 38
Overview & Logistics My hometown Jason Filippou (UMCP) Discrete Structures 05-31-2016 11 / 38
Overview & Logistics My home country Jason Filippou (UMCP) Discrete Structures 05-31-2016 12 / 38
Overview & Logistics My home country Jason Filippou (UMCP) Discrete Structures 05-31-2016 13 / 38
Subject of the course Subject of the course Jason Filippou (UMCP) Discrete Structures 05-31-2016 14 / 38
Subject of the course Discrete Mathematics: The big picture Jason Filippou (UMCP) Discrete Structures 05-31-2016 15 / 38
Subject of the course Discrete Mathematics: The big picture MATHEMATICS Induction Calculus Logic Prob-Stats DISCRETE CONTINUOUS Set theory Optimization Number Functional Theory Analysis Jason Filippou (UMCP) Discrete Structures 05-31-2016 16 / 38
Subject of the course Short history of Discrete Mathematics Short history of Discrete Mathematics Jason Filippou (UMCP) Discrete Structures 05-31-2016 17 / 38
Subject of the course Short history of Discrete Mathematics Historical Overview The history of Discrete Mathematics largely runs parallel to that of Logic and Set Theory . Logic: Ancient Greece, Medieval Middle East, 19th century “renaissance”. Set theory: Cantor’s and Dedekind’s set theory, Russel’s and Tarski’s paradoxes, G¨ odel’s Incompleteness Theorem Extensions in Computer Science Comput ability theory Comput ation theory Jason Filippou (UMCP) Discrete Structures 05-31-2016 18 / 38
Subject of the course Short history of Discrete Mathematics Ancient Greece Thales of Miletus: First philosopher to use deductive reasoning. Euclid : Defined axioms , propositions as well as the notion of a formal proof . Authored Elements , the first collection of axioms of geometry and number theory. Aristotle : Authored Organon , with which he tried to answer the questions: What constitutes a syllogism ? Which syllogisms are valid ?) Figure 3: Euclid Figure 4: Aristotle Jason Filippou (UMCP) Discrete Structures 05-31-2016 19 / 38
Subject of the course Short history of Discrete Mathematics Medieval Middle East Progress made on inductive (bottom-up) reasoning. Avicennian logic was the dominant paradigm. The principles of mathematical induction were laid down at that time. Figure 5: Ibn Sina Figure 6: Not Ibn Sina Jason Filippou (UMCP) Discrete Structures 05-31-2016 20 / 38
Subject of the course Short history of Discrete Mathematics Modern Era Rigorous formalization of Logic. Leibniz Boole Russell / Whitehead Peano Hilbert Applications to binary circuits after World War II Figure 7: George Boole Figure 8: Bertrand Russel Jason Filippou (UMCP) Discrete Structures 05-31-2016 21 / 38
Subject of the course Short history of Discrete Mathematics Set Theory Axiomatization of set theory in the late 19th century. Cantor & Dedekind ( Cantorian set theory). Russel’s Paradox. 1 Hilbert’s Hotel. Limitations of the algorithmic procedure. odel’s Incompleteness Theorem TM G¨ Tarski’s Undefinability Theorem TM The halting problem. Figure 10: Alfred Tarski Figure 9: Georg Cantor 1 Independently and simultaneously discovered by Ernst Zermelo. Jason Filippou (UMCP) Discrete Structures 05-31-2016 22 / 38
Subject of the course As a Computer Scientist... As a Computer Scientist... Jason Filippou (UMCP) Discrete Structures 05-31-2016 23 / 38
Subject of the course As a Computer Scientist... Where Discrete Math fits (in CS) Mathematical backbone for Theory! Counting and probability paramount! Inductive proofs of correctness everywhere. Applications of logic “Vanilla” logic, DataLog and deductive databases. Probabilistic logics (e.g MLNs) and graph databases. Automated theorem provers (commercial / academic prototypes) Set Theoretical elements paramount for: Computability theory. Study of compilers. Jason Filippou (UMCP) Discrete Structures 05-31-2016 24 / 38
Subject of the course As a Computer Scientist... But I just want to code! So you want to be hired by a software company. 4 to 5 interviews. First 2 questions in 1st - 2nd interviews are usually low-level theoretical and may contain examples such as: Among the residents of [insert name of city that you’re interviewing for a position at], is it possible that you can find two people with the exact same number of hairs on their head? If I have a full binary tree of height 10 and I add another level of leaves to it, how many nodes will I have total? First question is an application of the Pigeonhole Principle. Second question requires an inductive proof as a step in the answer (some might argue 2 inductive proofs, one classic and one structural). Jason Filippou (UMCP) Discrete Structures 05-31-2016 25 / 38
Subject of the course As a CS-UMD student... As a CS-UMD student... Jason Filippou (UMCP) Discrete Structures 05-31-2016 26 / 38
Subject of the course As a CS-UMD student... Where Discrete Structure fits (in the curriculum) 250 216 Jason Filippou (UMCP) Discrete Structures 05-31-2016 27 / 38
Subject of the course As a CS-UMD student... Where Discrete Structure fits (in the curriculum) 351 250 216 Jason Filippou (UMCP) Discrete Structures 05-31-2016 28 / 38
Subject of the course As a CS-UMD student... Where Discrete Structure fits (in the curriculum) (Also: 330, 351 320,...) 250 216 Jason Filippou (UMCP) Discrete Structures 05-31-2016 29 / 38
Subject of the course As a CS-UMD student... Where Discrete Structure fits (in the curriculum) 421 430 (Also: 330, 351 320,...) 250 216 Jason Filippou (UMCP) Discrete Structures 05-31-2016 30 / 38
What we’ll (tentatively) cover What we’ll (tentatively) cover Jason Filippou (UMCP) Discrete Structures 05-31-2016 31 / 38
What we’ll (tentatively) cover Part 1 Logic (Weeks 1 & 2). Propositional logic. Applications on Boolean Circuits. “Predicate” logic. Formal proof methodology (Weeks 2 & 3). Existential proofs. Constructive proofs. Proofs by contradiction. Jason Filippou (UMCP) Discrete Structures 05-31-2016 32 / 38
What we’ll (tentatively) cover 1 st Midterm! Friday, 06-17. In-class, 85 minutes. Jason Filippou (UMCP) Discrete Structures 05-31-2016 33 / 38
Recommend
More recommend
