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CSci 2011 Discrete Mathematics Lecture 1 Introduction

CSci 2011 Discrete Mathematics Lecture 1 Introduction. Yongdae Kim. Instructor, TA, Office Hours. Instructor Yongdae Kim ( Fourth time teaching 2011) Email:   kyd(at)cs. umn. edu Please include 2011 in the subject of your mail

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CSci 2011 Discrete Mathematics Lecture 1 Introduction

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  1. CSci 2011 Discrete MathematicsLecture 1Introduction Yongdae Kim CSci 2011

  2. Instructor, TA, Office Hours • Instructor • Yongdae Kim (Fourth time teaching 2011) • Email:   kyd(at)cs. umn. edu • Please include 2011 in the subject of your mail • Office: 200 Union St. SE, EECS Building (Keller Hall), room 4-225E • Office Hours: T 11:00 ~ 12:00, Th 10:00 ~ 11:00 (Also by appointment) • Teaching Assistants • Ben Dischinger, disch029(at)umn.edu, MW 9:45 AM - 10:45 AM • Abedelaziz Mohaisen (Aziz), mohaisen(at)cs.umn.edu, Th 11:30 AM - 1:30 PM • Jeremy Iverson, jiverson(at)cs.umn.edu, Tu 2:30 PM - 3:30 PM • Shaun Goss, goss0063(at)umn.edu, Th 2:30 PM - 3:30 PM • Nathan Fox, foxxx340(at)umn.edu, Th 1:30 PM - 2:30 PM • Katie Wolf, wolfx265(at)umn.edu, W 1:30 PM - 2:30 PM • Recitation • Section 001: Ben (Lead), Katie, Shaun • Section 002: Ben (Lead), Aziz, Katie • Section 003: Aziz (Lead), Ben, Nathan • Section 004: Aziz (Lead), Jeremy, Shaun • Section 005: Jeremy (Lead), Shaun, Nathan CSci 2011

  3. Class web page, e-mail • http://www-users.itlabs.umn.edu/classes/Fall-2010/csci2011/ • Reading the page carefully and regularly! • Read the Syllabus carefully. • Check calendar. • E-mail policy • Include [2011] in the subject of your e-mail • Use TA as much as possible :-) CSci 2011

  4. Textbook • Discrete Mathematics and Its Applications, • Rosen • 6th Edition • McGraw Hill • 2006 CSci 2011

  5. Overview • Much of the basic mathematical machinery useful in computer science will be presented, with applications. • Students will learn actively the art of creating real-world proofs in these areas, • preparing them for diverse regions of computer science such as architecture, algorithms, automata, programming languages, cryptography, and • increasing their general problem-solving abilities in all areas. CSci 2011

  6. Problems solved using Discrete Math • How many secure passwords? • Probability of winning Texas Hold’em? • How can I encrypt a message? • Shortest paths between two cities using public transportation? • How many steps required to sort 10,000 numbers? Is this algorithm correct? • How to design a circuit that multiply two integers? CSci 2011

  7. Why study Discrete Maths? • Proof • Ability to understand and create mathematical argument • Gateway to more advanced CS courses • Data structures, algorithms, automata theory, formal languages • Database, networks, operating system, security CSci 2011

  8. Guide for Successful Study • No minimalist approach • Homework would be sufficient! NOPE!!! • Read relevant sections before coming to class • Do the homework (of course!!!) • Solve much more problems (odd numbered) • Work regularly • Most chapters are building blocks for other chapters • So you cannot catch up 2 week lectures in 2 days • On average 10 hours EVERY week! • Creativity • No questions will require you to put just numbers to formula. • Need to know how to apply! This can be improved by practice! • Learning  Book, class, note, homework • It is combination of everything! • Think yourself, discuss with your friends, write your own answer! CSci 2011

  9. Course content very approximately in temporal order • Ch. 1: Logic and Proofs • Ch. 2: Sets, Functions, Sequences and Sums • Ch. 3: Algorithms, the Integers, and Matrices • Ch. 4: Induction and Recursion • Ch. 5: Counting • Ch. 6: Discrete Probability • Ch. 8: Relations • Ch. 12: Modeling Computation CSci 2011

  10. Typical Schedule • Tuesday • Lecture: 75 minutes • Group Work Due (Given in recitation section, every week) • Thursday • Lecture: 75 minutes • Assignment due (Every other week) • Posted on Sunday (1.5 week is given) • Topics covered until the Tuesday in the same week • Quiz: Every other Wednesday (50 min) • Wednesday • Recitation: 50 minutes • Group assignment. (Formed by instructor) • Due: next Tuesday CSci 2011

  11. Evaluation (IMPORTANT!) • The following rules will be strictly enforced. • Evaluation: • Assignments (6), group assignments (12), quizzes (6), and a Final exam. • You must pass every quiz individually by attaining at least 50% of the available points on each • Students who fail more than once will receive an F for the course. • All quizzes and examinations are closed book and closed notes. (One page cheat sheet is OK.) • Do not schedule any absence (especially on Thursday) during the semester - there are no make-up quizzes. CSci 2011

  12. Due dates and Submission • Due dates for all assignments are strict • All assignments must be received at the very start of the class to receive credit. • No late assignment will be graded. • Keep a copy of each of your submissions as evidence that you have indeed submitted each assignment. • Do not ever put your assignment under the instructor’s office door. CSci 2011

  13. Grading • Absolute (i.e. not on a curve). • The overall grade will be based upon • 3% for each homework, 1% for each group assignment, 7% for each quiz, and 28% for the final. • A minimum of 60% is necessary for an S or C- grade. • Grading will be as follows • 95.0% or above yields an A, 90.0% an A- • 85% = B+, 80% = B, 75% = B- • 70% = C+, 65% = C, 60% = C- • 55% = D+, 50% = D, and less than 50% yields an F. • Percentages are not rounded when using this scheme. • Extra credit questions will be always available. CSci 2011

  14. Grading questions and Complaints • Grading is performed by the TAs. • If you have a question about grading, talk to TAs. • Only if something unreasonable has occurred will the instructor intervene. • Furthermore, there is a limit of ten days from when an assignment or quiz is returned in recitation (whether you are there to receive it or not) for grading problems to be dealt with. • After that period, such will not be considered. • The sole exception to this rule is the final examination. CSci 2011

  15. And… • Incompletes (or make up exams) will in general not be given. • Exception: a provably serious family or personal emergency arises with proof and the student has already completed all but a small portion of the work. • Scholastic conduct must be acceptable. Specifically, you must do your assignments, quizzes and examinations yourself, on your own. CSci 2011

  16. Survey • Office Hour • Network and computer security research • Math, math, math… CSci 2011

  17. Propositions • A proposition is a statement that can be either true or false • “Yongdae has an Apple laptop.” • “Yongdae is a professor.” • “3 = 2 + 1” • “3 = 2 + 2” • Not propositions: • “Are you Bob?” • “x = 7” • “I am heavy.” CSci 2011

  18. Propositional variables • We use propositional variables to refer to propositions • Usually are lower case letters starting with p (i.e. p, q, r, s, etc.) • A propositional variable can have one of two values: true (T) or false (F) • A proposition can be… • A single variable: p • An operation of multiple variables: p(qr) CSci 2011

  19. Introduction to Logical Operators • About a dozen logical operators • Similar to algebraic operators + * - / • In the following examples, • p = “Today is Friday” • q = “Today is my birthday” CSci 2011

  20. Logical operators: Not • A “not” operation switches (negates) the truth value • Symbol:  or ~ • p = “Today is not Friday” CSci 2011

  21. Logical operators: And • An “and” operation is true if both operands are true • Symbol:  • It’s like the ‘A’ in And • pq = “Today is Friday and today is my birthday” CSci 2011

  22. Logical operators: Or • An “or” operation is true if either operands are true • Symbol:  • pq = “Today is Friday or today is my birthday (or possibly both)” CSci 2011

  23. Logical operators: Conditional 1 • A conditional means “if p then q” • Symbol:  • pq = “If today is Friday, then today is my birthday” • p→q=¬pq the antecedent the consequence CSci 2011

  24. Logical operators: Conditional 2 • Let p = “I am elected” and q = “I will lower taxes” • I state: p  q = “If I am elected, then I will lower taxes” • Consider all possibilities • Note that if p is false, then the conditional is true regardless of whether q is true or false CSci 2011

  25. Logical operators: Conditional 3 • Alternate ways of stating a conditional: • p implies q • If p, q • p only if q • p is sufficient for q • q if p • q whenever p • q is necessary for p CSci 2011

  26. Logical operators: Conditional 4 CSci 2011

  27. Logical operators: Bi-conditional 1 • A bi-conditional means “p if and only if q” • Symbol:  • Alternatively, it means “(if p then q) and (if q then p)” • Note that a bi-conditional has the opposite truth values of the exclusive or CSci 2011

  28. Logical operators: Bi-conditional 2 • Let p = “You take this class” and q = “You get a grade” • Then pq means “You take this class if and only if you get a grade” • Alternatively, it means “If you take this class, then you get a grade and if you get a grade then you take (took) this class” CSci 2011

  29. Boolean operators summary • Learn what they mean, don’t just memorize the table! CSci 2011

  30. Precedence of operators • Just as in algebra, operators have precedence • 4+3*2 = 4+(3*2), not (4+3)*2 • Precedence order (from highest to lowest): ¬   → ↔ • The first three are the most important • This means that p  q  ¬r→s↔tyields: (p  (q  (¬r)) →s) ↔ (t) • Not is always performed before any other operation CSci 2011

  31. Translating English Sentences • Question 7 from Rosen, p. 17 • p = “It is below freezing” • q = “It is snowing” • It is below freezing and it is snowing • It is below freezing but not snowing • It is not below freezing and it is not snowing • It is either snowing or below freezing (or both) • If it is below freezing, it is also snowing • It is either below freezing or it is snowing, but it is not snowing if it is below freezing • That it is below freezing is necessary and sufficient for it to be snowing pq p¬q ¬p¬q pq p→q ((pq)¬(pq))(p→¬q) p↔q CSci 2011

  32. Translation Example 2 • Heard on the radio: • A study showed that there was a correlation between the more children ate dinners with their families and lower rate of substance abuse by those children • Announcer conclusions: • If children eat more meals with their family, they will have lower substance abuse • If they have a higher substance abuse rate, then they did not eat more meals with their family CSci 2011

  33. Translation Example 3 • “I have neither given nor received help on this exam” • Let p = “I have given help on this exam” • Let q = “I have received help on this exam” • ¬p¬q CSci 2011

  34. Translation Example 4 • You can access the Internet from campus only if you are a computer science major or you are not a freshman. • a  (c  f) • You cannot ride the roller coaster if you are under 4 feet tall unless you are older than 16 years old. • (f  s)  r • r  ( f  s) CSci 2011

  35. Boolean Searches (2011 OR 5471) AND yongdae AND “computer science” • Note that Google requires you to capitalize Boolean operators • Google defaults to AND; many others do not CSci 2011

  36. Bit Operations • Boolean values can be represented as 1 (true) and 0 (false) • A bit string is a series of Boolean values. Length of the string is the number of bits. • 10110100 is eight Boolean values in one string • We can then do operations on these Boolean strings • Each column is its ownBoolean operation 01011010 10110100 11101110 CSci 2011

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