1. Week 9 - Wednesday

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

4.  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.

6.  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.  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.  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.  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] = 

11.  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 1. n | (a – b) 2. a  b (mod n) 3. a = b + kn for some integer k 4. a and b have the same remainder when divided by n 5. a mod n = b mod n

12.  Let a, b, c, d and n be integers with n > 1  Let a  c (mod n) and b  d (mod n), then: 1. (a + b)  (c + d) (mod n) 2. (a – b)  (c – d) (mod n) 3. ab  cd (mod n) 4. a m  c m (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 -1 a  1 (mod n)  I'd love to have us learn how to find this, but there isn't time

14.  Let R be a relation on a set A  R is antisymmetric iff 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 R antisymmetric?  Let S be the "divides" relation on the set of all integers  Is S antisymmetric?

15.  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.  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.  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?

19.  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.  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.  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.  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.  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

25.  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 A A B B B B B

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

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

28.  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.  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.  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.  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.  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.  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?

35.  Read Chapter 9

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