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Math/CSE 1019N: Discrete Mathematics for Computer Science Winter 2007

Math/CSE 1019N: Discrete Mathematics for Computer Science Winter 2007. Suprakash D at ta datta@cs.yorku.ca Office: CSEB 3043 Phone: 416-736-2100 ext 77875 Course page: http://www.cs.yorku.ca/course/1019. Administrivia. Textbook:. Lectures: Tu-Th 10:00-11:30 am (CLH E)

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Math/CSE 1019N: Discrete Mathematics for Computer Science Winter 2007

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  1. Math/CSE 1019N:Discrete Mathematics for Computer ScienceWinter 2007 Suprakash Datta datta@cs.yorku.ca Office: CSEB 3043 Phone: 416-736-2100 ext 77875 Course page: http://www.cs.yorku.ca/course/1019

  2. Administrivia Textbook: Lectures: Tu-Th 10:00-11:30 am (CLH E) Exams: 2 tests (40%), final (40%) Homework (20%): equally divided between 3 or 4 assignments. Slides: should be available the previous evening Office hours: Tuesday 12-2 pm, Wed 1-2 pm or by appointment at CSEB 3043. Kenneth H. Rosen. Discrete Mathematics and Its Applications, Sixth Edition. McGraw Hill, 2007.

  3. Administrivia – contd. • Cheating will not be tolerated. Visit the webpage for more details on policies. • TA: Tutorials/office hours TBA. • I will have some extra-credit quizzes. These will be announced beforehand. • HW submitted late will not be graded.

  4. Course objectives We will focus on two major goals: • Basic tools and techniques in discrete mathematics • Propositional logic • Set Theory • Simple algorithms • Induction, recursion • Counting techniques (Combinatorics) • Introductory graph theory • Precise and rigorous mathematical reasoning • Writing proofs

  5. To do well you should: • Study with pen and paper • Ask for help immediately • Practice, practice, practice… • Follow along in class rather than take notes • Ask questions in class • Keep up with the class • Read the book, not just the slides

  6. Today • Ch. 1: Introduction to Propositional Logic • Truth values, truth tables • Boolean logic:    • Implications:  

  7. Why study propositional logic? • A formal mathematical “language” for precise reasoning. • Start with propositions. • Add other constructs like negation, conjunction, disjunction, implication etc. • All of these are based on ideas we use daily to reason about things.

  8. Propositions • Declarative sentence • Must be either True or False. Propositions: • York University is in Toronto • York University is in downtown Toronto • All students at York are Computer Sc. majors. Not propositions: • Do you like this class? • There are x students in this class.

  9. Propositions - 2 • Truth value: True or False • Variables: p,q,r,s,… • Negation: • p (“not p”) • Truth tables

  10. Conjunction, Disjunction • Conjunction: p q [“and”] • Disjunction: p  q [“or”]

  11. Exclusive OR (XOR) • p  q – T if p and q have different truth values, F otherwise • Colloquially, we often use OR ambiguously – “an entrée comes with soup or salad” implies XOR, but “students can take MATH XXXX if they gave taken MATH 2320 or MATH 1019” usually means the normal OR (so a student who has taken both is still eligible for MATH XXXX).

  12. Conditional • p  q [“if p then q”] • p: hypothesis, q: conclusion • E.g.: “If you turn in a homework late, it will not be graded”; “If you get 100% in this course, you will get an A+”. • TRICKY: Is p  q TRUE if p is FALSE? YES!! • Think of “If you get 100% in this course, you will get an A+” as a promise – is the promise violated if someone gets 50% and does not receive an A+?

  13. Conditional - 2 • p  q [“if p then q”] • Truth table: Note the truth table of p  q

  14. Logical Equivalence • p  q and p  q are logically equivalent • Truth tables are the simplest way to prove such facts. • We will learn other ways later.

  15. Contrapositive • Contrapositive of p  q isq p • Any conditional and its contrapositive are logically equivalent (have the same truth table) – Check by writing down the truth table. • E.g. The contrapositive of “If you get 100% in this course, you will get an A+” is “If you do not get an A+ in this course, you did not get 100%”.

  16. Converse • Converse of p  q isq  p • Not logically equivalent to conditional • Ex 1:“If you get 100% in this course, you will get an A+” and “If you get an A+ in this course, you scored 100%” are not equivalent. • Ex 2: If you won the lottery, you are rich.

  17. Other conditionals Inverse: • inverse of p  q isp q • How is this related to the converse? Biconditional: • “If and only if” • True if p,q have same truth values, flase otherwise. Q: How is this related to XOR? • Can also be defined as (p  q)  (q  p)

  18. Readings and notes • Read pages 1-12. • Think about the notion of truth tables. • Master the rationale behind the definition of conditionals. • Practice translating English sentences to propositional logic statements.

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