Discrete Mathematics

1. Discrete Mathematics 0. INTRODUCTION TO DISCRETE MATHEMATICS Lecture 1 Dr.-Ing. Erwin Sitompul http://zitompul.wordpress.com

2. Text book: Kenneth H. Rosen, “Discrete Mathematics and Its Applications”, 6th Edition, McGraw-Hill International Edition, 2007. Text Book and Syllabus • Syllabus: • Logic and Proofs • Sets • Relation and Functions • Mathematical Induction • Counting and Combinatorial • Graphs

3. Grade Policy Final Grade = 10% Homework + 20% Quizzes + 30% Midterm Exam + 40% Final Exam + Extra Points • Homeworks will be given in fairly regular basis. The average of homework grades contributes 10% of final grade. • Homeworks must be written on A4 papers. • Homeworks must be submitted on time. If you submit late, < 10 min.  No penalty 10 – 60 min.  –20 points > 60 min.  –40 points • There will be 3 quizzes along the semester. The average of the best 2 will be counted. The quizes contribute 30% of final grade. • Midterm and final exam schedule will be announced in time. • Make up of quizzes and exams will be held one week after the schedule of the respective quizzes and exams.

4. Lecture Material • New lecture slides will be available on internet every week. Please check the course homepage regularly. • The course homepage is : • http://zitompul.wordpress.com • You are responsible to read and understand the lecture slides. If there is any problem, you may ask me. • Quizzes, midterm exam, and final exam will be closed-book. But you may bring an A5 cheat sheet for quizzes and A4 cheat sheet for mid and final exam. • Extra points will be given if you solve a problem in front of the class. You will earn 1, 2, or 3 points.

5. What is “Discrete Mathematics”? Discrete Mathematics is a branch of mathematics that discuss about discrete objects or components. What is the definition of discrete? • An object can be said to be discrete if : • It consists of unconnected distinct parts/ members • It consists of finite or countable parts/ members. • Example: • A set of female IR 2009 students • A set of positive integers

6. What is “Discrete Mathematics”? The opposite of discrete is continuous. All engineers take mathematics through Calculus and differential equations, but IE is different in that it is based on “discrete variable” math, whereas all other engineering is based on “continuous variable” math. The emphasis on discrete mathematics, along with linear algebra and difference equation, distinguishes Industrial Engineering from other engineering disciplines.

7. Sequencing orders Scheduling batches Determining the number of materials handling units Arranging factory layouts Finding sequences of motions etc. What is “Discrete Mathematics”? Discrete Mathematics can find unique applications in IE on Production Systems such as:

8. What is “Discrete Mathematics”? • Some examples of problems related with discrete mathematics problems: • How many different password can be made out of 8 different characters? • How does a ISBN number of a book or a credit card number is validated? • How many 8-bit-long binary string combinations can be made if the sum of bit-1 must be odd? • How to determine the shortest path between point A and point B in a factory complex? • Proof that a combination of 3s and 5s can result any integer number higher than 8.

9. What is “Discrete Mathematics”? • Some examples of problems related with discrete mathematics problems: • How to construct logic circuit for a seven segments? • Can you walk through all the streets in your housing complex exactly once and come back to the original position? • “Cheap food is not tasty.”“Tasty food is not cheap.” Are both statements telling us the same thing?

10. Discrete Mathematics 1. LOGIC AND PROOFS Lecture 1 Dr.-Ing. Erwin Sitompul http://zitompul.wordpress.com

11. Logic and Proposition • Logic: • Logic is the basic of reasoning. • Reasoning is based on relations between propositions. • Proposition: • A proposition is a statement that can be either true (T) or false (F), but not both. • The synonym of proposition is open sentence.

12. Proposition • Example: (Propositions) • 13 is an odd number. • Ir. Soekarno was graduated from UGM. • 1 + 1 = 2. • 8  square root of (8 + 8). • There is monkey in the moon. • Today is Wednesday. • For any integer n 0, there exists 2n which is an even number. • x + y = y + x for any real number x and y.

13. Proposition • Example: (Not Propositions) • What time does Argo Bromo train arrive at Gambir Station? • Do the quiz without cooperating! • x = 3 + 8. • x > 5. • I am heavy. Conclusion: Propositions are declarative sentences. Conclusion: If a proposition is made out of mathematical equations, then the equations must posses an answer so that its truth value can be evaluated.

14. Proposition Propositions are denoted with lower case letters starting with p such as p, q, r, … • Example: • p : 13 is an odd number. • q : Ir. Soekarno was graduated from UGM. • r : 2 + 2 = 4.

15. Combining Propositions • Given propositions p and q, then: • Negation of p : not p, p, ¬p • Conjunction : p and q, pq • Disjunction : p or q, pq Consisting only a single statement, each p and q is called atomic proposition. Combination of p and q results in a compound proposition.

16. Combining Propositions Example: The following prepositions are known p : Today is rainy. q : The class is cancelled. pq : Today is rainy andthe class is cancelled. pq : Today is rainy orthe class is cancelled. p : It is not truethat today is rainy.(or: Today is not rainy)

17. Combining Propositions Example: Given the following propositions, p : The girl is beautiful. q : The girl is smart. Express the following proposition combinations using symbolic notation. The girl is beautiful and smart. The girl is beautiful but not smart. The girl is neither beautiful nor smart. It is not true that the girl is ugly or not smart. The girl is beautiful, or ugly and smart. That the girl is ugly as well as smart, is not true. pq pq pq (pq) p(pq) (pq)

18. T F F Truth Table Negation Conjunction Disjunction Example: p : 17 is a prime number. q : Prime number is always odd. pq : 17 is a prime number and prime number is always odd.

19. Proposition Operation in Google

20. Compound Proposition Example: Build the truth table of the proposition (pq)  (qr).

21. Tautology and Contradiction Compound proposition which is always true for all cases is called tautology. Compound proposition which is always false for all cases is called contradiction. Example: p (pq) is a tautology

22. Tautology and Contradiction Example: (pq)  (pq) is a contradiction

23. Equivalence of Compound Proposition Two compound proposition P(p,q,…) and Q(p,q,…) are said to be logicallyequivalent if they have identical truth table. Notation: P(p,q,…) Q(p,q,…) Example: De Morgan’s Law (pq)  pq

24. Logical Laws

25. Logical Laws

26. Logical Equivalence Example: Show that p ~(pq) and p ~q are logically equivalent. p ~(pq ) p (~p ~q) (De Morgan’s Law)  (p ~p)  (p ~q) (Distributive Law)  T  (p ~q) (Negation Law) p ~q (Identity Law)

27. Logical Equivalence Example: Proof the truth of Absorption Lawp (pq) p p (pq)  (p F)  (pq) (Identity Law) p (F q) (Distributive Law) p F (Domination Law) p (Identity Law)

28. Exclusive Disjunction • The word “or” can be used in two ways in logical operation: • Inclusive or“or” in the sense “p or q or either”Example: “The required IE engineer must master AutoCAD or SolidWorks”. • Exclusive or“or” in the sense “p or q but not both”Example: “He is punished with 5 years prison plus Rp 500 millions fine or additional 3 months prison.

29. Exclusive Disjunction Logical operator for exclusive disjunction is xor, with the notation .

30. Conditional Proposition Conditional proposition is also called implication. • Form: “If p, then q” • Notation: pq • Proposition p is called hypothesis, antecedent, premise, or condition. • Proposition q is called conclusion or consequence.

31. Conditional Proposition • Example: • If I pass the exams, then I will get presents from my parents. • If the temperature reaches 80 C, then the alarm will buzz. • If you have not enrolled properly, then your name will not be printed on the attendance list. Truth table for implication:

32. Conditional Proposition Lecturer: “If your final exam grade is 80 or more, then you will get an A for this subject.” Case 1: Your final exam grade is higher than 80 (true hypothesis) and you get an A for the subject (true conclusion). The lecturer tells the truth.TRUE Case 2: Your final exam grade is higher than 80 (true hypothesis)but you do not get an A (false conclusion). The lecturer tells a lie.FALSE Case 3: Your final exam grade is lower than 80 (false hypothesis) and you get an A (true conclusion). The lecturer cannot be said to be wrong or telling a lie.Maybe he/she see your extra efforts and high motivation and thus without any doubt to give you an A. TRUE Case 4: Your final exam grade is lower than 80 (false hypothesis) and you do not get an A (false conclusion). The lecturer tells the truth.TRUE

33. Conditional Proposition • Various ways to express implication pq: • If p, then q. • If p, q. • p implies/causes q. • q if p. • p only if q. • p is the sufficient condition for q. • p is sufficient for q. • q is the necessary condition for p. • q is necessary for p. • q whenever p.

34. Conditional Proposition • Example: Implication in various forms • If today is rainy, then the flowers will grow well. • If the gas pedal is pressed deeper, the car moves fast. • Ice melted in north and south pole causes the increase of sea level. • He is willing to go if he is given travel allowance. • Ahmad can take Factory Layout only if he already passed Discrete Mathematics. • The sufficient condition for a gas station to explode is small cigarette sparks. • The necessary condition for Indonesia to win the WorldCupTM is by hiring a famous foreign trainer. !! Convert all propositions above into “If p then q”

35. Conditional Proposition Example: Show that pq is logically equivalent with ~pq. “If p, then q”  “Not p or q” Example: Determine the negation of pq. ~(pq)  ~(~pq)  ~(~p)  ~q  p ~q

36. Conditional Propositions Express the following proposition combinations using symbolic notation. You can access the internet from campus only if you are a student or you are not a university guest. You cannot ride the roller coaster if you are under 100 cm tall unless you are older than 16 years old. a (s¬g) (u¬o) ¬r

37. Homework 1 Two merchants publish new marketing campaign to attract more customers. The first merchant launches a motto “Good stuffs are not cheap”, while the second merchants says “Cheap stuffs is not good”. a) Examine whether both mottos tell the same message or not. b) In your opinion, which motto is better?

38. Homework 1 Examine whether both following statements are logically equivalent or not. S1: Excellent physical condition is the necessary condition to become a soldier. S2: If someone does not have excellent physical condition, then he cannot become a soldier. New