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Lecture 1.1: Course Overview, and Propositional Logic*

This lecture provides an overview of the CS 250 course, including administrative details, the instructor's background, prerequisites, expectations, and grading policies. It also introduces propositional logic as a fundamental topic in discrete mathematics.

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Lecture 1.1: Course Overview, and Propositional Logic*

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  1. Lecture 1.1: Course Overview, and Propositional Logic* CS 250, Discrete Structures, Fall 2011 Nitesh Saxena *Adopted from previous lectures by Cinda Heeren

  2. Outline • Administrative Material • Introductory Technical Material Lecture 1 - Course Overview, and Propositional Logic

  3. Some Important Pointers • Instructor: Nitesh Saxena • Office: CH 133 • Email: saxena@cis.uab.edu (best way to reach me!) • Phone No: 205-975-3432 • Office Hours: Thursdays 3-4pm (or by appointment) • Course Web Page (also accessible through my web-page) http://www.cis.uab.edu/saxena/teaching/cs250-f11/ • TA: • Song Gao: gaos@uab.edu • Ph.D. student; research in data mining • Office Hours: Mondays and Wednesdays 1-2pm • Blackboard: https://cms.blazernet.uab.edu/cgi-bin/bb9login Lecture 1 - Course Overview, and Propositional Logic

  4. About the Instructor • PhD graduate from UC Irvine • Previously an Assistant Professor at the Polytechnic Institute of New York University • Research in computer and network security, and applied cryptography • Web page: http://cis.uab.edu/saxena Lecture 1 - Course Overview, and Propositional Logic

  5. Prerequisites • Fundamental course for anyone intending to become a computer scientist • MA 106 (pre-calculus trigonometry) OR • MA 107 (pre-cal algebra/trigonometery) OR • MA 125 (calculus I) OR • MA 126 (calculus II) OR • MA 227 (calculus III) • Minimum grade of C Lecture 1 - Course Overview, and Propositional Logic

  6. What to expect • The course would be quite involved and technical • Lot of mathematics • Busy schedule • The grading will be curved • I would love to give all A’s but I won’t mind giving F’s when deserved  • I might/will make mistakes • Please point them out • Talk to me if you have any complaints (or send me an anonymous email ) • But, I guarantee that • I will encourage you to do your best • You’ll have fun • I’ll help you learn as much as I can – don’t hesitate to ask for help whenever needed Lecture 1 - Course Overview, and Propositional Logic

  7. What I expect of you • Please do attend lectures • Review lecture slides after each lecture • Solve text book exercises as you read through the chapters • Ask questions in the class • Ask questions over email • Attend office hours • Try to start early on homework assignments • Don’t wait until the very last minute! • Follow the instructions and submit assignments on time Lecture 1 - Course Overview, and Propositional Logic

  8. Course Textbook • Discrete Mathematics and its Applications  -- Kenneth Rosen • Seventh Edition Lecture 1 - Course Overview, and Propositional Logic

  9. Grading • 40% - 6 homework assignments • 60% - Exams • 2 mid-terms: 30% (15% each) • 1 final: 30% Lecture 1 - Course Overview, and Propositional Logic

  10. Policies Against Cheating or Misconduct • You are not allowed to collaborate with any other student, in any form, while doing your homeworks, unless stated otherwise; perpetrators will at least fail the course or disciplinary action may be taken • No collaboration of any form is allowed on exams • You can definitely refer to online materials and other textbooks; but whenever you do, you should cite so in your homeworks. This is a rule of thumb. • Also check: http://main.uab.edu/Sites/undergraduate-programs/general-studies/academic-success/67537/ Lecture 1 - Course Overview, and Propositional Logic

  11. Late Homework Policy • None – no late homeworks are allowed • Either you submit on time and your homework will be graded OR you submit late and the homework is NOT graded • You should stick to deadlines, please • Exception will be made ONLY under genuine circumstances Lecture 1 - Course Overview, and Propositional Logic

  12. Tentative Course Schedule (29 lectures) • Logic and Proofs (Chap 1) – 5 lectures • Propositional Logic (1.1, 1.2) • Equivalences (1.3) • Quantifiers and Predicates (1.4, 1.5) • Proof Techniques (1.7. 1.8) • Basic Structures (Chap 2) – 5 lectures • Sets and Set Operations (2.1, 2.2) • Functions (2.3) • Sequences and Summations (2.4) • Matrices (2.6) • Induction and Recursion (Chap 5) – 6 lectures • Induction (5.1) • Strong Induction and Well Ordering (5.2) • Recursion and Structural Induction (5.3) • Recursive Algorithms (5.4) • Program Correctness (5.5) • Relations (Chap 9) – 5 lectures • Relations and Properties (9.1) • Closures and Equivalence (9.4, 9.5) • Partial Orderings (9.6) • Graphs (Chap 10) – 5 lectures • Graphs, Terminologies, and Models(10.1, 10.2) • Isomorphism and Connectivity (10.3, 10.4) • Paths and Shortest Path Problem (10.5, 10.6) • Planar Graphs and Coloring (10.7, 10.8) • Miscellaneous, if time permits – 3 lectures • Counting (Chap 6) • Trees (Chap 11) • Number Theory (Chap 4) Lecture 1 - Course Overview, and Propositional Logic

  13. Scheduled Travel • Sept 17-21 • Attending ACM Conference on Ubiquitous Computing, Beijing, China • We will have our Mid-Term 1 on Sept 20, tentatively • Likely no lecture missed due to this travel • Oct 17-21 • Attending ACM Conference on Computer and Communications Security, Chicago • We will have our Mid-Term 2 during this week, tentatively • 1 lecture will be missed – perhaps to be covered by a guest lecturer. However, this will not affect our overall course schedule and topic coverage. Lecture 1 - Course Overview, and Propositional Logic

  14. Instructions • HW submissions • Name your files “Lastname_Firstname_HW#” • Submit it on Blackboard • Please make sure that you have correctly submitted/uploaded the files (simply “saving” them may not be sufficient) • PDF format only • Check the course web-site regularly • I am posting lecture slides and homeworks there • Check your UAB email regularly • I am sending out announcements there • e.g., when I post homeworks • Only use your UAB email to communicate with me and the TA • NO EXCUSES for not following instructions Lecture 1 - Course Overview, and Propositional Logic

  15. Propositional Logic Lecture 1 - Course Overview, and Propositional Logic

  16. Proposition • A proposition is a logical statement that is either TRUE (T) or FALSE (F) • Propositions (examples) • 3+2=32 • 3+2=5 • MA 106 is a prerequisite for CS 250 • It is sunny today • Not Propositions (examples) • What time it is now? • X+1 = 2 • Read this carefully • What grade can I get in this course? Lecture 1 - Course Overview, and Propositional Logic

  17. Notice that p is a proposition! PropositionalLogic-- Negation Suppose p is a proposition. The negation of p is written p and has meaning: “It is not the case that p.” In English, it is referred to as a “NOT” • Ex. “CS173 is NOT Bryan’s favorite class” is a negation for “CS173 is Bryan’s favorite class” Truth table for negation: Lecture 1 - Course Overview, and Propositional Logic

  18. Propositional Logic -- Conjunction Conjunction corresponds to English “AND”. p  q is true exactly when p and q are both true. • Ex. “Amy is curious and clever” is a conjunction of “Amy is curious” and “Amy is clever”. Truth table for conjunction: Lecture 1 - Course Overview, and Propositional Logic

  19. Propositional Logic -- Disjunction Disjunction corresponds to English “OR” p  q is true when p or q (or both) are true. It is actually an “inclusive OR” • Ex. “Michael is brave OR nuts” is a disjunction of “Michael is brave” and “Michael is nuts”. Truth table for disjunction: Lecture 1 - Course Overview, and Propositional Logic

  20. Propositional Logic – Exclusive OR p q is true when only one of p or q is true Ex: Students who have taken calculus or computer science, but not both, can enroll for this class. Truth table for xor: Lecture 1 - Course Overview, and Propositional Logic

  21. Propositional Logic – Implication Implication: pq corresponds to English “if p then q,” or “p implies q.” • If it is raining then it is cloudy • If you have taken MA 106, you can enroll for CS 250 • If you work hard then you can get a good grade Truth table for implication: Lecture 1 - Course Overview, and Propositional Logic

  22. Propositional Logic – Biconditional • This is equivalent to: (p  q)  (q  p) • Also, referred to as the “iff” condition • For p  q to be true, p and q must have the same truth value. Truth table for biconditional: Lecture 1 - Course Overview, and Propositional Logic

  23. Complex Composite Propositionsand Equivalence • Combination of many propositions using different operations (negation, conjunction, disjunction, implication) • Precedence order for these operations: • Negation • Conjunction • Disjunction • Implies • Biconditional • A complex proposition can often be reduced to a simple one • This means that the complex proposition and the simple proposition are logically equivalent Lecture 1 - Course Overview, and Propositional Logic

  24. Propositional Logic – Logical Equivalence • p is logically equivalent to q if their truth tables are the same. We write p q. • In other words, p is logically equivalent to q if p  q is True. • We will study about equivalences more in the next lecture • But, for now, let us look at some examples Lecture 1 - Course Overview, and Propositional Logic

  25. Propositional Logic – Logical Equivalence Challenge: Try to find a proposition that is equivalent to pq, but that uses only the connectives , , and . Lecture 1 - Course Overview, and Propositional Logic

  26. Distributive Law – an example of equivalence Distributivity: p (qr)  (p q)  (pr) Lecture 1 - Course Overview, and Propositional Logic

  27. One of these things is not like the others. Hint: In one instance, the pair of propositions is equivalent. pq q  p Propositional Logic – special definitions Contrapositives: pq and q  p • Ex. “If it is noon, then I am hungry.” “If I am not hungry, then it is not noon.” Converses: pq andq  p • Ex. “If it is noon, then I am hungry.” “If I am hungry, then it is noon.” Inverses: pq and p  q • Ex. “If it is noon, then I am hungry.” “If it is not noon, then I am not hungry.” Lecture 1 - Course Overview, and Propositional Logic

  28. Propositional Logic – special definitions A tautology is a proposition that’s always TRUE. A contradiction is a proposition that’s always FALSE. T T F F Lecture 1 - Course Overview, and Propositional Logic

  29. Propositional Logic – bit-wise operators • All the operators are extensible and applicable to “bits” and “bitstrings” • ‘1’ is TRUE and ‘0’ is FALSE Lecture 1 - Course Overview, and Propositional Logic

  30. Propositional Logic – applications • Computer Programs • Propositional logic is a key to writing good code…you can’t do any kind of conditional (if) statement without understanding the condition you’re testing. • Different programming languages may have different syntax for logic operators • Hardware and Gates: • All the logical connectives we’ve discussed are also found in hardware and are called “gates.” • Foundational Element for Proof Systems and Proof Techniques • Ex: Classical proofs in provable cryptography based on counterpositives. • Logical searches • Ex: “Alabama” and “Universities” • Ex: “java – coffee” • Writing Policies • Ex: firewall policies • Logical Puzzles and Games • … Lecture 1 - Course Overview, and Propositional Logic

  31. Some Quick Questions • Is “what is your name” a proposition? • Is “The sun revolves around the earth” a proposition? If so, what is its logical value. • What is the negation of “The sun revolves around the earth”? What is the logical value of this negation. • “You can get a good grade if you perform well on homeworks and you perform well on exams” – represent as a proposition. • “You may fail the course if you cheat or you do not attend any lectures” – represent as a proposition. • What is p  p equivalent to? • What is p  p equivalent to? • What is: • 1  1? • 0  1? • 1 0? Lecture 1 - Course Overview, and Propositional Logic

  32. Some Quick Questions • If p is True and q is False, what is p -> q? • Is p->q equivalent to p  q? • What will be the output of the following piece of pseudocode: X = 50; If ( X > 35) print “Pass”; Else print “Fail”; • How would (p  q)  (r s) be represented in computer hardware (using gates)? • What is the converse of: “if you are a good student, you will end up getting a good grade”? Lecture 1 - Course Overview, and Propositional Logic

  33. Today’s Reading and Next Lecture • Rosen 1.1 and 1.2, and part of 1.3 • Please start solving the exercises at the end of each chapter section. They are fun. • Please read 1.3 and 1.4 in preparation for the next lecture Lecture 1 - Course Overview, and Propositional Logic

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