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CS 173: Discrete Mathematical Structures

CS 173: Discrete Mathematical Structures. Cinda Heeren heeren@cs.uiuc.edu Rm 2213 Siebel Center Ofc Hr: Monday 9-11a. CS 173 Announcements. This is where you’ll usually see announcements Today is a special day…every announcement gets its own slide. CS 173 Class and sections. Lecture:

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CS 173: Discrete Mathematical Structures

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  1. CS 173:Discrete Mathematical Structures Cinda Heeren heeren@cs.uiuc.edu Rm 2213 Siebel Center Ofc Hr: Monday 9-11a

  2. CS 173 Announcements • This is where you’ll usually see announcements • Today is a special day…every announcement gets its own slide Cs173 - Spring 2004

  3. CS 173 Class and sections • Lecture: • 9:45a-10:15a, Siebel 1404 • 11a-12:15p, Siebel 1103 • Section: • One hour, once a week • Sign up is mandatory, but not attendance • http://compass. uiuc.edu bluestem -> select spring 07 -> select “cs173 sections” (can only choose a section once) Cs173 - Spring 2004

  4. I reserve the right to be more lenient. CS 173 Grading • Every activity in class gives you a certain number of points: • Section attendance: 1 pt/wk (12 total) • Homework: 2 pt/wk (24 total) • 2 midterms, 20 points each (40 points total) • 1 final exam, 30 points (30 points total) • Grading scale: • A: 88+ • B: 76 - 88 • C: 64 - 76 • D: 52 - 64 Cs173 - Spring 2004

  5. CS 173 Homework • Details: • Assigned Monday, due Sunday, 8a (electronic submission) • No late hwk accepted. • Graded by your section leader and returned in section. • Formatting instructions on web site. • Hwk #1 available! Due 1/21, 8a. • Collaboration policy: PLEASE work together, form study groups, get consumed. BUT write your solutions independently. Cs173 - Spring 2004

  6. CS 173 Exams • Midterms - 2 evening exams (7-9p): • 3/1 (lecture cancelled that day) • 4/5 (lecture cancelled that day) • Location TBD and announced on web site nearer the exam • Conflicts will be accommodated (send email to Cinda) • Academic integrity (applies also to Hwk): • http://www.admin.uiuc.edu/policy/code/article_1/a1_1-401.html Cs173 - Spring 2004

  7. CS 173 Miscellaneous • Text: Rosen 5th or 6th ed. • Course web page: http://www.cs.uiuc.edu/class/cs173 • Jabber chat room: https://agora.cs.uiuc.edu/display/tsg/Jabber+Instant+Messaging+Service • Class wiki: https://agora.cs.uiuc.edu/display/cs173/Home Cs173 - Spring 2004

  8. CS 173 Expectations • This is a FUN course! • This class contains some of the most beautiful topics you’ll ever learn. • It’s even useful, particularly to your CS experience. Hints for success: • Read the textbook • Lectures really do help • Use the problem sessions, newsgroup, and study groups • Do the homework assignments! Cs173 - Spring 2004

  9. Table entries: # of queries ‘til everyone knows CS 173 What’s that on my head? • In each group of players there is at least one red card. • The game is over when everyone knows the color of the card on his/her head. • Inquisitor repeatedly queries “do you know what’s on your head?” • Puzzle: how many queries until everyone knows? # red cards Group size Cs173 - Spring 2004

  10. CS 173 Discrete Mathematical Structures????? Calculus - the study of things that change continuously. • How long does it take to walk from here to SC? Discrete Math - the study of things that can be counted. • How many students are in this class? • How many typos are on these slides? Math techniques particular to discrete objects: • Counting and combinatorics • Graph representation • Probability • Logic Cs173 - Spring 2004

  11. CS 173 Computer Science? This sounds like a math course! • No programming assignments • Lots of proofs and paper/pencil solutions THIS is the math underlying most of CS. Computer Scientists rely on this kind of math to: • Design high speed networks • Design interactive games • Design speech recognition programs • Create web search algorithms • … Cs173 - Spring 2004

  12. If this class has a greater purpose, this is it! CS 173 Course Focus Language of Discrete Math • Logic and set theory • Relations and functions • Counting and combinatorics • Graphs and trees Tools for proving things • Direct methods • Induction and recursion Algorithm analysis • Running time and efficiency Cs173 - Spring 2004

  13. CS 173 Propositional Logic What’s a proposition? A proposition is a declarative statement that’s either TRUE or FALSE (but not both). Cs173 - Spring 2004

  14. CS 173 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.” • Ex. It is not the case that cs173 is Eric’s favorite class. Truth table for negation: Cs173 - Spring 2004

  15. CS 173 Propositional Logic - more operators Conjunction: p  q corresponds to English “and.” Proposition p  q is true when p and q are both true. • Ex. Katie is brave and clever. Truth table for conjunction: Cs173 - Spring 2004

  16. CS 173 Propositional Logic - more operators Disjunction: p  q corresponds to English “or.” Proposition p  q is true when p or q (or both) is true. • Ex. Katie is brave or nuts. Truth table for disjunction: Cs173 - Spring 2004

  17. CS 173 Propositional Logic - more operators Exclusive Or: p  q corresponds to English “either…or…” Proposition p  q is true when p or q (not both) is true. • Ex. You will eat either pizza or a burrito for lunch. Truth table for exclusive or: F T T F Cs173 - Spring 2004

  18. CS 173 Propositional Logic - more operators Implication: p  q corresponds to English “if…then…” • Ex. If it is raining, then it is cloudy. • Ex. If there are 200 students in the room then I am the Easter Bunny. • Ex. If I win the lottery then I will give you $5. Truth table for implication: T F T T Cs173 - Spring 2004

  19. CS 173 Propositional Logic - more operators Biconditional: p  q corresponds to English “if and only if…then…” • Ex. If and only if it is raining, then it is cloudy. • Ex. If and only if there are 200 students in the room then I am the Easter Bunny. Truth table for biconditional: T F F T Cs173 - Spring 2004

  20. CS 173 Propositional Logic - more operators? • How many of these binary operators could we do? • How many of them do we need? (HUH?) To answer, we need a notion of “logical equivalence.” If p and q are propositions, then p q if their truth tables are the same. If p q, we say “p is equivalent to q.” Cs173 - Spring 2004

  21. CS 173 Propositional Logic - logical equivalence Challenge: Try to find a proposition that is equivalent to pq, but that uses only the connectives , , and . Cs173 - Spring 2004

  22. One of these things is not like the others. Hint: In one instance, the pair of propositions is equivalent. pq q  p CS 173 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.” Cs173 - Spring 2004

  23. YES NO NO CS 173 Propositional Logic - special  • Contrapositives: p  q  q  p ? Ex. “If it is noon, then I am hungry.” “If I am not hungry, then it is not noon.” • Converses: p  q  q  p ? Ex.“If it is noon, then I am hungry.” “If I am not hungry, then it is not noon.” • Inverses: p  q  p  q ? Ex.“If it is noon, then I am hungry.” “If it is not noon, then I am not hungry.” Cs173 - Spring 2004

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