1. Week 8 - Monday

2.  What did we talk about last time?  Properties of functions  One-to-one  Onto  Inverses  Cardinality

4.  An Arab sheikh tells his two sons to race their camels to a distant city to see who will inherit his fortune  The one whose camel is slower wins  After wandering aimlessly for days, the brothers ask a wise man for guidance  Upon receiving the advice, they jump on the camels and race to the city as fast as they can  What did the wise man say to them?

5.  Consider f(x) = 3x 2 – 9 on R  Does it have an inverse?  What if its domain were restricted to R + ?  Consider f(x) = 8x + 17 on R  Does it have an inverse?  What is it?

7.  Cardinality gives the number of things in a set  Cardinality is:  Reflexive: A has the same cardinality as A  Symmetric: If A has the same cardinality as B, B has the same cardinality as A  Transitive: If A has the same cardinality as B, and B has the same cardinality as C, A has the same cardinality as C  For finite sets, we could just count the things to determine if two sets have the same cardinality

8.  Because we can't just count the number of things in infinite sets, we need a more general definition  For any sets A and B, A has the same cardinality as B iff there is a bijective mapping A to B  Thus, for any element in A, it corresponds to exactly one element in B, and everything in B has exactly one corresponding element in A

9.  Show that the set of positive integers has the same cardinality as the set of all integers  Hint: Create a bijective function from all integers to positive integers  Hint 2: Map the positive integers to even integers and the negative integers to odd integers

10.  A set is called countably infinite if it has the same cardinality as Z +  You have just shown that Z is countable  It turns out that (positive) rational numbers are countable too, because we can construct a table of their values and move diagonally across it, numbering values, skipping numbers that have been listed already 1/11/21/31/4 2/12/22/32/4 3/13/23/33/4 4/14/24/34/4

11.  We showed that positive rational numbers were countable, but a trick similar to the one for integers can show that all rational numbers are countable  The book gives a classic proof that real numbers are not countable, but we don't have time to go through it  For future reference, the cardinality of positive integers, countable infinity, is named  0 (pronounced aleph null)  The cardinality of real numbers, the first uncountable infinity (because there are infinitely many uncountable infinities), is named  1 (pronounced aleph 1)

12. Student Lecture

14.  Relations are generalizations of functions  In a function, an element of the domain must map to exactly one element of the co-domain  In a relation, an element from one set can be related to any number (from zero up to infinity) of other elements  Like functions, we're usually going to focus on binary relations  We can define any binary relation between sets A and B as a subset of A x B

15.  For binary relation R,  x R y  (x, y)  R  Let R be a relation from Z to Z such that (x, y)  R iff x – y is even  Is 1 R 3?  Is 2 R 3?  Is 2 R 2?  Let C be a relation from R to R such that (x, y)  R iff x 2 + y 2 = 1  Is (1, 0)  C?  Is 0 C 0?  Is (-1/2,  3/2)  C?

16.  Consider the sets A = {2, 4, 6} and B = {1, 3, 5}  Let R be the relation {(2,5), (4,1), (4,3), (6,5)}  Draw the arrow diagram for R  Is R a function?  Let S be the relation for all (x, y)  A x B, (x, y)  S iff y = x + 1  Draw the arrow diagram for S  Is S a function?  x 2 + y 2 = 1 on real numbers is not a function for both reasons

17.  We've relaxed things considerably by moving from functions to relations  All relations have inverses (just reverse the order of the ordered pairs)  Example  Let A = {2,3,4} and B = {2,6,8}  For all (x, y)  A x B, x R y  x | y  List the ordered pairs in R  List the ordered pairs in R -1

18.  A directed graph describes a relationship between nodes  One way to record a graph is as a matrix  We can also think of a directed graph as a relation from a set to itself  What's the relation for this directed graph? a a c c d d b b e e

20.  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

21.  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

22.  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

23.  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?

24.  If you have a set A and a binary relation R on A, the transitive closure of R called R t satisfies the following properties:  R t is transitive R  RtR  Rt  If S is any other transitive relation that contains R, then R t  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 R t

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

26.  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?

28.  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

29.  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

30.  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)

31.  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] = 

33.  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

34.  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

36.  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?

37.  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

40.  Finish relations  Review for Exam 2

41.  Work on Homework 6  Due Friday at exam  Study for Exam 2  Friday in class  Read Chapter 9