Week 9 - Wednesday. What did we talk about last time? Exam 2 Before that: review Before that:...

CS322Week 9 - Wednesday

Last time

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

Questions?

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.

Equivalence Relations

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

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

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)

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] =

Modular Arithmetic

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 equivalent1. n | (a – b)2. a b (mod n)3. a = b + kn for some integer k4. a and b have the same remainder when divided by

n5. a mod n = b mod n

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:

1. (a + b) (c + d) (mod n)2. (a – b) (c – d) (mod n)3. ab cd (mod n)4. 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

Partial Orders

Antisymmetry

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?

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

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

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?

Probability

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)

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?

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?

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?

? ? ?

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

Multiplication Rule

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

A

A

B

B

B

B

B

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)

Coin example

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

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?

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

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?

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?

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)!

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?

Upcoming

Next time…

Read Chapter 9

Reminders

Work on Homework 7 Due on Friday

Summer internship opportunity at Masonic Villages Contact me if interested