1. September1999 CMSC 203 / 0201 Fall 2002 Week #1 – 28/30 August 2002 Prof. Marie desJardins

2. September1999 October 1999 TOPICS  Course overview  Propositional logic  LaTeX

3. September1999 October 1999 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. September1999 October 1999 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. September1999 October 1999 Course overview and policies  Overview ..\index.html..\index.html  Academic integrity  Homework and grading  Exams  Class participation

6. September1999 October 1999 Tools  Emacs  LaTeX  Maple  Class mailing list  Class website

7. September1999 WED 8/28 PROPOSITIONAL LOGIC

8. September1999 October 1999 CONCEPTS / VOCABULARY  Propositions  Truth value  Truth table  Operators: Negation, conjunction, disjunction, exclusive or, implication, XOR, biconditional  Converse, inverse, contrapositive

9. September1999 October 1999 Examples  Truth table for XOR (Table 1.1.4)  Truth table for biconditional (Table 1.1.6)

10. September1999 October 1999 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. September1999 October 1999 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. September1999 October 1999 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. September1999 October 1999 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. September1999 FRIDAY 8/30 LaTeX, PROPOSITIONAL EQUIVALENCES

15. September1999 October 1999 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. September1999 October 1999 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. September1999 October 1999 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. September1999 October 1999 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. September1999 October 1999 FOR NEXT TIME  Reading: Ch. 1.3  Homework 0 due at (or before) the beginning of the next class!