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CMSC 203 / 0201 Fall 2002

CMSC 203 / 0201 Fall 2002. Week #1 – 28/30 August 2002 Prof. Marie desJardins. TOPICS. Course overview Propositional logic LaTeX. What’s discrete math?. Mathematics of integers and collections of objects Main topics covered in course: Logic and sets Sequences and summations

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CMSC 203 / 0201 Fall 2002

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  1. CMSC 203 / 0201Fall 2002 Week #1 – 28/30 August 2002 Prof. Marie desJardins

  2. TOPICS • Course overview • Propositional logic • LaTeX

  3. What’s discrete math? • Mathematics of integers and collections of objects • Main topics covered in course: • Logic and sets • Sequences and summations • Number theory: Integers, matrices • Algorithms: Analysis, recursion, correctness • Counting: Probability, permutations, combinations • Relations • Machines: Finite-state and Turing

  4. Thinking mathematically • Formalize an informally worded problem • Apply problem-solving techniques to find a solution • Use formal proof techniques to demonstrate the correctness of your solution • Present the proof clearly and readably

  5. Course overview and policies • Overview • ..\index.html • Academic integrity • Homework and grading • Exams • Class participation

  6. Tools • Emacs • LaTeX • Maple • Class mailing list • Class website

  7. WED 8/28PROPOSITIONAL LOGIC

  8. CONCEPTS / VOCABULARY • Propositions • Truth value • Truth table • Operators: Negation, conjunction, disjunction, exclusive or, implication, XOR, biconditional • Converse, inverse, contrapositive

  9. Examples • Truth table for XOR (Table 1.1.4) • Truth table for biconditional (Table 1.1.6)

  10. Examples II • Exercise 1.1.7 • p = “You drive over 65 miles per hour” • q = “You get a speeding ticket” • (a) You do not drive over 65 miles per hour. • (b) You drive over 65 mph, but you do not get a speeding ticket. • (c) You will get a speeding ticket if you drive over 65 mph. • (d) If you do not drive over 65 mph, then you will not get a speeding ticket. • (e) Driving over 65 mph is sufficient for getting a speeding ticket. • (f) You get a speeding ticket, but you do not drive over 65 mph. • (g) Whenever you get a speeding ticket, you are driving over 65 mph.

  11. Examples III • Exercise 1.1.11: Disjunction vs. exclusive or: What are the two interpretations? Which is intended? • “To take discrete mathematics, you must have taken calculus or a course in computer science.” • “When you buy a new car from Acme Motor Company, you get $2000 back in cash or a 2% car loan.” • “Dinner for two includes two items from column A or three items from column B.” • “School is closed if more than 2 feet of snow falls or if the wind chill is below -100.”

  12. Examples IV • Exercise 1.1.21: Construct truth tables for the following compound propositions. • (a) p  p • (c) (p  q)  q • (e) (p  q)  (q  p)

  13. FOR NEXT TIME • Chapter 1.1-1.2; also look at 1.3 if you have time • Handouts: • Survey (DUE FRIDAY 8/30) • Academic integrity/grading policy (DUE FRIDAY 8/30) • LaTeX handout • Week 1 slides • Homework #0 (DUE WEDNESDAY 9/4)

  14. FRIDAY 8/30 LaTeX, PROPOSITIONAL EQUIVALENCES

  15. CONCEPTS / VOCABULARY • Tautology, contradiction • Table 1.2.5 (p. 17) of logical equivalences: • Identity laws • Domination laws • Idempotent laws • Double negation law • Commutative laws • Associative laws • Distributive laws • De Morgan’s laws

  16. Examples • Prove De Morgan’s Laws with truth tables (Example 1.2.2) •  (p  q)  p  q •  (p  q)  p  q • Prove with logical equivalences: • Equivalence of a proposition and its contrapositive • Equivalence of a proposition’s inverse and its converse • From Exercise 1.1.21(e): • Prove that (p  q)  ( q   p) is a tautology, using logical equivalences • Last time, we showed that this is a tautology using a truth table

  17. Examples II • Exercise 1.2.9: Prove using logical equivalences • (a) (p  q)  p • (b) p  (p  q) • (c) p  (p  q) • (d) (p  q)  (p  q) • (e) (p  q)  p • (f) (p  q)  q

  18. Examples III • Exercises 1.2.30-33, 35 • p NAND q (equivalently, p | q) is true iff either p or q or both are false • p NOR q (equivalently, p  q) is true iff both p and q are false • 1.2.30: Construct a truth table for NAND • 1.2.31: Show that p | q is logically equivalent to (p  q) • 1.2.32: Construct a truth table for NOR • 1.2.33: Show that p  q is logically equivalent to (p  q) • 1.2.35: Find a proposition equivalent to p → q using only the logical operator 

  19. FOR NEXT TIME • Reading: Ch. 1.3 • Homework 0 due at (or before) the beginning of the next class!

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