CS322Week 8 - Wednesday

Last time

What did we talk about last time? Cardinality Countability Relations

Questions?

Logical warmup

There is a Grand Hotel with an infinite number of rooms

Their slogan is: "We're always full, but we always have room for you!" Although completely full, when a single guest arrives, he or

she can be accommodated by moving into Room 1 The guest in Room 1 will move into Room 2, the guest in

Room 2 will move into Room 3, and so on, with each guest in Room n moving into Room n + 1

Unfortunately, tragedy has struck! The hotel across the street, Inn Finite, which also has a (countably) infinite number of rooms, has just burned down

Describe how all of Inn Finite's guests can be accommodated at the Grand Hotel

Quick SOLHRRwCC review

To solve sequence ak = Aak-1 + Bak-2

Find its characteristic equation t2 – At – B = 0

If the equation has two distinct roots r and s Substitute a0 and a1 into an = Crn + Dsn

to find C and D If the equation has a single root r

Substitute a0 and a1 into an = Crn + Dnrn to find C and D

Reflexive, Symmetric, and Transitive PropertiesStudent Lecture

Properties of Relations

Reflexive

We will discuss many useful properties that hold for any binary relation R on a set A (that is from elements of A to other elements of A)

Relation R is reflexive iff for all x A, (x, x) R

Informally, R is reflexive if every element is related to itself

R is not reflexive if there is an x A, such that (x, x) R

Symmetric

Relation R is symmetric iff for all x, y A, if (x, y) R then (y, x) R

Informally, R is symmetric if for every element related to another element, the second element is also related to the first

R is not symmetric if there is an x, y A, such that (x, y) R but (y, x) R

Transitive

Relation R is transitive iff for all x, y, z A, if (x, y) R and (y, z) R then (x, z) R

Informally, R is transitive when an element is related to a second element and a second element is related to a third, then it must be the case that the first element is also related to the third

R is not transitive if there is an x, y, z A, such that (x, y) R and (y, z) R but (x, z) R

Examples

Let A = {0, 1, 2, 3} Let R = {(0,0), (0,1), (0,3), (1,0), (1,1), (2,2), (3,0),

(3,3)} Is R reflexive? Is R symmetric? Is R transitive?

Let S = {(0,0), (0,2), (0,3), (2,3)} Is S reflexive? Is S symmetric? Is S transitive?

Let T = {(0,1), (2,3)} Is T reflexive? Is T symmetric? Is T transitive?

Transitive closures

If you have a set A and a binary relation R on A, the transitive closure of R called Rt satisfies the following properties: Rt is transitive R Rt

If S is any other transitive relation that contains R, then Rt S

Basically, the transitive closure just means adding in the least amount of stuff to R to make it transitive

If you can get there in R, you can get there directly in Rt

Transitive closure example

Let A = {0, 1, 2, 3} Let R = {(0,1), (1,2), (2,3)} Is R transitive? Then, find the transitive closure of R

Examples on real sets

Let R be a relation on real numbers R such that x R y x = y Is R reflexive? Is R symmetric? Is R transitive?

Let S be a relation on real numbers R such that x S y x < y Is S reflexive? Is S symmetric? Is S transitive?

Let T be a relation on positive integers N such that m T n 3 | (m – n) Is T reflexive? Is T symmetric? Is T transitive?

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

Quiz

Upcoming

Next time…

Review for Exam 2

Reminders

Work on Homework 6 Due Friday before midnight

Study for Exam 2 Next Monday in class Review on Friday