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1. Week 9 - Wednesday CS322

2. Last time • What did we talk about last time? • Exam 2 • Before that: review • Before that: relations

3. Questions?

4. Logical warmup • This is an old one, and not especially mathematical • Still, it illustrates a useful point • A man and his son are driving in a car one day, when they get into a terrible accident • The man is killed instantly • The boy is knocked unconscious, but he is still alive • He is rushed to a hospital, and will need immediate surgery • The doctor enters the emergency room, looks at the boy, and says, "I can't operate on this boy, he is my son." • How can this be? • Please be quiet if you have heard this one before.

5. Equivalence Relations

6. Partitions • A partition of a set A (as we discussed earlier) is a collection of nonempty, mutually disjoint sets, whose union is A • A relation can be induced by a partition • For example, let A = {0, 1, 2, 3, 4} • Let A be partitioned into {0, 3, 4}, {1}, {2} • The binary relation induced by the partition is: x R y x and y are in the same subset of the partition • List the ordered pairs in R

7. Equivalence relations • Given set A with a partition • Let R be the relation induced by the partition • Then, R is reflexive, symmetric, and transitive • As it turns out, any relation R is that is reflexive, symmetric, and transitive induces a partition • We call a relation with these three properties an equivalence relation

8. Congruences • We say that m is congruent to n modulo d if and only if d | (m – n) • We write this: • m n (mod d) • Congruence mod d defines an equivalence relation • Reflexive, because m m (mod d) • Symmetric because m n (mod d) means that n m (mod d) • Transitive because m n (mod d) and n k (mod d) mean that m k (mod d) • Which of the following are true? • 12  7 (mod 5) • 6  -8 (mod 4) • 3  3 (mod 7)

9. Equivalence classes • Let A be a set and R be an equivalence relation on A • For each element a in A, the equivalence class of a, written [a], is the set of all elements x in A such that a R x • Example • Let A be { 0, 1, 2, 3, 4, 5, 6, 7, 8} • Let R be congruence mod 3 • What's the equivalence class of 1? • For A with R as an equivalence relation on A • If b [a], then [a] = [b] • If b [a], then [a]  [b] = 

10. Modular Arithmetic

11. Modular arithmetic • Modular arithmetic has many applications • For those of you in Security, you know how many of them apply to cryptography • To help us, the following statements for integers a, b, and n, with n > 1, are all equivalent • n | (a – b) • a b (mod n) • a = b + kn for some integer k • a and b have the same remainder when divided by n • a mod n = b mod n

12. Rules of modular arithmetic • Let a, b, c, d and n be integers with n > 1 • Let a c (mod n) and b  d (mod n), then: • (a + b)  (c + d) (mod n) • (a – b)  (c – d) (mod n) • ab cd (mod n) • am cm (mod n), for all positive integers m • If a and n are relatively prime (share no common factors), then there is a multiplicative inverse a-1 such that a-1a  1 (mod n) • I'd love to have us learn how to find this, but there isn't time

13. Partial Orders

14. Antisymmetry • Let R be a relation on a set A • R is antisymmetriciff for all a and b in A, if a R b and b R a, then a = b • That is, if two different elements are related to each other, then the relation is not antisymmetric • Let R be the "divides" relation on the set of all positive integers • Is Rantisymmetric? • Let S be the "divides" relation on the set of all integers • Is Santisymmetric?

15. Partial orders • A relation that is reflexive, antisymmetric, and transitive is called a partial order • The subset relation is a partial order • Show it's reflexive • Show it's antisymmetric • Show it's transitive • The less than or equal to relation is a partial order • Show it's reflexive • Show it's antisymmetric • Show it's transitive

16. Hasse Diagrams • Let set A = {1, 2, 3, 9, 18} • Let R be the "divides" relation on A • Draw A as a set of points and connect each pair of points with arrows if they are related with R • Now, delete all loops and transitive arrows • This is a Hasse Diagram

17. Total orders • Let R be a partial order on set A • Elements a,b R are comparable if either a R b or b R a (or both) • If all the elements in a partial order are comparable, then the partial order is a total order • Let R be the "less than or equal to" relation on R • Is it a total order? • Let S be the "divides" relation on positive integers • Is it a total order?

18. Probability

19. Definitions • A sample space is the set of all possible outcomes • An event is a subset of the sample space • Formula for equally likely probabilities: • Let S be a finite sample space in which all outcomes are equally likely and E is an event in S • Let N(X) be the number of elements in set X • Many people use the notation |X| instead • The probability of E is P(E) = N(E)/N(S)

20. Card examples • There are 52 cards in a normal Anglo-American deck of cards • Four suits: Spades, Hearts, Clubs, and Diamonds • 13 denominations: Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King • Imagine you draw a single card from a thoroughly shuffled deck • What is the sample space? • What is the event of drawing a black face card? • What is the probability of drawing a black face card?

21. Dice example • Six-sided dice have, uh, six sides, numbered 1 through 6 • If you roll two dice • What is the sample space of outcomes? • What is the event that the two dice add up to 7? • What is the probability that the two dice add up to 7? • What about all the other possible values?

22. Monty Hall • Imagine you are playing a game show with 3 doors • There is a prize behind one and nothing behind the other two • As the contestant, you pick a door, but it isn't opened yet • The host Monty Hall opens one of the other two doors, revealing nothing • Then, you get a chance to switch • Should you stay or switch or does it matter? ? ? ?

23. Counting the elements in a list • As a computer scientist, you have almost certainly figured this out • But, just to formalize it, if you have a list numbered m through n, with no elements missing, the total number of elements are n – m + 1 • For example, there are 50 elements in an array indexed from 0 to 49

24. Multiplication Rule

25. Possibility trees • We can use a tree to represent all the possibilities in a situation • Example: • Teams A and B are playing a best of 3 tournament • The first team to win 2 games wins • How likely is it that 3 games are needed to decide the tournament, assuming that all ways of playing the tournament are equally likely? A A A B B A B A B B

26. Multiplication rule • If an operation has k steps such that • Step 1 can be performed in n1 ways • Step 2 can be performed in n2 ways … • Step k can be performed in nk ways • Then, the entire operation can be performed in n1n2 … nk ways • This rule only applies when each step always takes the same number of ways (unlike the previous possibility tree example)

27. Coin example • If you flip a coin k times, how many total possibilities are there for the outcomes?

28. Personal Identification Numbers • If a PIN is a 4 digit sequence, where each digit is 0-9 or A-Z, how many PINs are possible? • How many PINs are possible if no digits are repeated? • Assuming that all PINs are equally likely, what's the probability that a PIN chosen at random has no repetitions?

29. Permutations • A permutation of a set of objects is an ordering of the objects in a row • Consider set { a, b, c } • Its permutations are: • abc • acb • cba • bac • bca • cab • If a set has n 1 elements, it has n! permutations

30. Permutations of letters in a word • How many different ways can the letters in the word "WOMBAT" be permuted? • How many different ways can "WOMBAT" be permuted such that "BA" remains together? • What is the probability that, given a random permutation of "WOMBAT", the "BA" is together? • How many different ways can the letters in "MISSISSIPPI" be permuted? • How many would it be if we don't distinguish between copies of letters?

31. Permuting around a circle • What if you want to seat 6 people around a circular table? • If you only care about who sits next to whom (rather than who is actually in Seat 1, Seat 2, etc.) how many circular permutations are there? • What about for n people?

32. Permutations of selected elements • An r-permutation of a set of n element is an ordered selection of r elements from the set • Example: A 2-permutation of {a, b, c} includes: • ab • ac • ba • bc • ca • cb • The number of r-permutations of a set of n elements is P(n,r) = n!/(n – r)!

33. r-permutation examples • What is P(5,2)? • How many 4-permutations are there in a set of 7 objects? • How many different ways can three of the letters in "BYTES" be written in a row?

34. Upcoming

35. Next time… • Read Chapter 9

36. Reminders • Work on Homework 7 • Due on Friday • Summer internship opportunity at Masonic Villages • Contact me if interested