Lecture 1.1: Course Overview, and Propositional Logic CS 250, Discrete Structures, Fall 2013 Nitesh Saxena Adopted from previous lectures by Cinda Heeren

10/14/2015 Lecture 1 - Course Overview, and Propositional Logic2 Outline Administrative Material Introductory Technical Material

10/14/2015 Lecture 1 - Course Overview, and Propositional Logic3 Some Important Pointers Instructor: Nitesh Saxena Office: CH (best way to reach Phone No: Office Hours: Thursdays 3-4pm (or by appointment) Course Web Page (also accessible through my web- page) TA/Grader: Lutfor Rahman: M.S. Office location: Ugrad lab (CH 154) Office Hours: Mondays and Wednesdays 12:30-1:30pm Blackboard:

10/14/2015 Lecture 1 - Course Overview, and Propositional Logic4 About the Instructor Associate Professor, CIS 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: Research group (SPIES):

10/14/2015 Lecture 1 - Course Overview, and Propositional Logic5 Prerequisites 1. Fundamental course for anyone intending to become a computer scientist 2. MA 106 (pre-calculus trigonometry) OR MA MA 107 (pre-cal algebra/trigonometery) OR MA MA 125 (calculus I) OR MA MA 126 (calculus II) OR MA MA 227 (calculus III) MA Minimum grade of C

10/14/2015 Lecture 1 - Course Overview, and Propositional Logic6 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 ) 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

7 What I expect of you Please do attend lectures Take notes Review lecture slides after each lecture Solve text book exercises as you read through the chapters Ask questions in the class Ask questions over 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

10/14/2015 Lecture 1 - Course Overview, and Propositional Logic8 Course Textbook Discrete Mathematics and its Applications -- Kenneth Rosen Seventh Edition

10/14/2015 Lecture 1 - Course Overview, and Propositional Logic9 Grading 40% - 5 homework assignments HW due every days or so 60% - Exams 2 mid-terms: 30% (15% each) 1 final: 30% Mid-Term 1 – Oct 1 st or 2 nd week (tentative) Mid-Term 2 – Nov 1 st or 2 nd week (tentative) Final – Dec 10

10/14/2015 Lecture 1 - Course Overview, and Propositional Logic10 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: code code

10/14/2015 Lecture 1 - Course Overview, and Propositional Logic11 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

10/14/2015 Lecture 1 - Course Overview, and Propositional Logic12 Tentative Course Schedule 1. Logic and Proofs (Chap 1) – ~5 lectures 1. Propositional Logic (1.1, 1.2) 2. Equivalences (1.3) 3. Quantifiers and Predicates (1.4, 1.5) 4. Proof Techniques ( ) 2. Basic Structures (Chap 2) – ~5 lectures 1. Sets and Set Operations (2.1, 2.2) 2. Functions (2.3) 3. Sequences and Summations (2.4) 4. Matrices (2.6) 3. Induction and Recursion (Chap 5) – ~6 lectures 1. Induction (5.1) 2. Strong Induction and Well Ordering (5.2) 3. Recursion and Structural Induction (5.3) 4. Recursive Algorithms (5.4) 5. Program Correctness (5.5) 4. Relations (Chap 9) – ~5 lectures 1. Relations and Properties (9.1) 2. Closures and Equivalence (9.4, 9.5) 3. Partial Orderings (9.6) 5. Graphs (Chap 10) – ~5 lectures 1. Graphs, Terminologies, and Models(10.1, 10.2) 2. Isomorphism and Connectivity (10.3, 10.4) 3. Paths and Shortest Path Problem (10.5, 10.6) 4. Planar Graphs and Coloring (10.7, 10.8) 6. Miscellaneous, if time permits – ~4 lectures 1. Counting (Chap 6) 2. Trees (Chap 11) 3. Number Theory (Chap 4)

10/14/2015 Lecture 1 - Course Overview, and Propositional Logic13 Scheduled Travel Usually conference and invited talks travel Usually no class the week of travel However, this will not affect our overall course schedule and topic coverage (perhaps a guest lecturer will cover on my behalf) Information about any travel will be provided as it becomes available

10/14/2015 Lecture 1 - Course Overview, and Propositional Logic14 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 regularly I am sending out announcements there e.g., when I post homeworks Only use your UAB to communicate with me and the TA Please specify CS 250 to subject line so I can identify course NO EXCUSES for not following instructions

10/14/2015 Lecture 1 - Course Overview, and Propositional Logic15 Propositional Logic

10/14/2015 Lecture 1 - Course Overview, and Propositional Logic16 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?

10/14/2015 Lecture 1 - Course Overview, and Propositional Logic17 Propositional Logic -- 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: p pp TFTF FTFT Notice that  p is a proposition!

10/14/2015 Lecture 1 - Course Overview, and Propositional Logic18 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: pqp  q TTFFTTFF TFTFTFTF TFFFTFFF

10/14/2015 Lecture 1 - Course Overview, and Propositional Logic19 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: pqp  q TTFFTTFF TFTFTFTF TTTFTTTF

10/14/2015 Lecture 1 - Course Overview, and Propositional Logic20 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: pqp q TTFFTTFF TFTFTFTF FTTFFTTF

10/14/2015 Lecture 1 - Course Overview, and Propositional Logic21 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: pqp  q TTFFTTFF TFTFTFTF TFTTTFTT

10/14/2015 Lecture 1 - Course Overview, and Propositional Logic22 Propositional Logic – Biconditional pqp  q TTFFTTFF TFTFTFTF TFFTTFFT 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:

10/14/2015 Lecture 1 - Course Overview, and Propositional Logic23 Complex Composite Propositions and Equivalence Combination of many propositions using different operations (negation, conjunction, disjunction, implication) Precedence order for these operations: 1. Negation 2. Conjunction 3. Disjunction 4. Implies 5. 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

10/14/2015 Lecture 1 - Course Overview, and Propositional Logic24 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

10/14/2015 Lecture 1 - Course Overview, and Propositional Logic25 Propositional Logic – Logical Equivalence Challenge: Try to find a proposition that is equivalent to p  q, but that uses only the connectives , , and . pqp  q TTFFTTFF TFTFTFTF TFTTTFTT pq  pq   p TTFFTTFF TFTFTFTF FFTTFFTT TFTTTFTT

10/14/2015 Lecture 1 - Course Overview, and Propositional Logic26 Distributive Law – an example of equivalence Distributivity: p  (q  r)  (p  q)  (p  r) pqr q  rq  rp  (q  r)p  qp  qp  rp  r(p  q)  (p  r) TTTTTTTT TTFFTTTT TFTFTTTT TFFFTTTT FTTTTTTT FTFFFTFF FFTFFFTF FFFFFFFF

10/14/2015 Lecture 1 - Course Overview, and Propositional Logic27 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 and q  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.” One of these things is not like the others. Hint: In one instance, the pair of propositions is equivalent. p  q   q   p

10/14/2015 Lecture 1 - Course Overview, and Propositional Logic28 Propositional Logic – special definitions A tautology is a proposition that’s always TRUE. A contradiction is a proposition that’s always FALSE. p ppp  pp  pp  pp  p TF FT TTTT FFFF

10/14/2015 Lecture 1 - Course Overview, and Propositional Logic29 Propositional Logic – bit-wise operators All the operators are extensible and applicable to “bits” and “bitstrings” ‘1’ is TRUE and ‘0’ is FALSE

10/14/2015 Lecture 1 - Course Overview, and Propositional Logic30 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 …

10/14/2015 Lecture 1 - Course Overview, and Propositional Logic31 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?

10/14/2015 Lecture 1 - Course Overview, and Propositional Logic32 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”?

10/14/2015 Lecture 1 - Course Overview, and Propositional Logic33 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