1. Week 8 - Wednesday

2.  What did we talk about last time?  Relations  Properties of relations  Reflexive  Symmetric  Transitive

4.  There are five pirates who wish to divide 100 gold coins  These pirates are ranked: 1. Captain 2. Lieutenant 3. Master 4. Midshipman 5. Seaman  In order of rank, each pirate gets the opportunity to propose a plan for dividing up the gold  If at least half of the pirates (including the proposer) agree on the proposition, it is carried out  Otherwise, the pirate is killed and the next highest ranking pirate makes a proposal  Pirates are completely rational, who value, in this order: 1. Staying alive 2. Maximizing gold coins received 3. Seeing other pirates die  If you were the captain, what would you propose?  Hint: Work backwards!

5.  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

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

7.  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?

9.  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

10.  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

11.  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)

12.  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] = 

14.  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

15.  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

17.  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?

18.  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

21.  In a proof by contradiction, you begin by assuming the negation of the conclusion  Then, you show that doing so leads to a logical impossibility  Thus, the assumption must be false and the conclusion true

22.  A proof by contradiction is different from a direct proof because you are trying to get to a point where things don't make sense  You should always mark such proofs clearly  Start your proof with the words Proof by contradiction  Write Negation of conclusion as the justification for the negated conclusion  Clearly mark the line when you have both p and ~p as a contradiction  Finally, state the conclusion with its justification as the contradiction found before

24.  Mathematical sequences can be represented in expanded form or with explicit formulas  Examples:  2, 5, 10, 17, 26, …  a i = i 2 + 1, i ≥ 1  Summation notation is used to describe a summation of some part of a sequence  Product notation is used to describe a product of some part of a sequence

25.  To prove a statement of the following form:   n  Z, where n  a, property P(n) is true  Use the following steps: 1. Basis Step: Show that the property is true for P(a) 2. Induction Step: ▪ Suppose that the property is true for some n = k, where k  Z, k  a ▪ Now, show that, with that assumption, the property is also true for k + 1

26.  Prove that, for all integers n  1,

27.  Using recursive definitions to generate sequences  Writing a recursive definition based on a sequence  Using mathematical induction to show that a recursive definition and an explicit definition are equivalent

28.  Expand the recursion repeatedly without combining like terms  Find a pattern in the expansions  When appropriate, employ formulas to simplify the pattern  Geometric series: 1 + r + r 2 + … + r n = (r n+1 – 1)/(r – 1)  Arithmetic series: 1 + 2 + 3 + … + n = n(n+ 1)/2

30.  To solve sequence a k = Aa k-1 + Ba k-2  Find its characteristic equation t 2 – At – B = 0  If the equation has two distinct roots r and s  Substitute a 0 and a 1 into a n = Cr n + Ds n to find C and D  If the equation has a single root r  Substitute a 0 and a 1 into a n = Cr n + Dnr n to find C and D  There will be one of these on the exam

32.  Defining finite and infinite sets  Definitions of:  Subset  Proper subset  Set equality  Set operations:  Union  Intersection  Difference  Complement  The empty set  Partitions  Cartesian product

33.  Proving a subset relation  Element method: Assume an element is in one set and show that it must be in the other set  Algebraic laws of set theory: Using the algebraic laws of set theory (given on the next slide), we can show that two sets are equal  Disproving a universal statement requires a counterexample with specific sets

34. NameLawDual Commutative A  B = B  AA  B = B  A Associative (A  B)  C = A  (B  C)(A  B)  C = A  (B  C) Distributive A  (B  C) = (A  B)  (A  C)A  (B  C) = (A  B)  (A  C) Identity A   = AA  U = A Complement A  A c = UA  A c =  Double Complement(A c ) c = A Idempotent A  A = AA  A = A Universal Bound A  U = UA   =  De Morgan’s (A  B) c = A c  B c (A  B) c = A c  B c Absorption A  (A  B) = AA  (A  B) = A Complements of U and  U c =  c = U Set Difference A – B = A  B c

35.  Use the element method to prove the following:  For all sets A, B, and C, if A ⊆ B then A ∩ C ⊆ B ∩ C

36.  It is possible to give a description for a set which describes a set that does not actually exist  For a well-defined set, we should be able to say whether or not a given element is or is not a member  If we can find an element that must be in a specific set and must not be in a specific set, that set is not well defined  Watch out for definitions that are logically inconsistent!

37.  Definitions  Domain  Co-domain  Range  Inverse image  Arrow diagrams  Poorly defined functions  Function equality

38.  One-to-one (injective) functions  Onto (surjective) functions  If a function F: X  Y is both one-to-one and onto (bijective), then there is an inverse function F -1 : Y  X such that:  F -1 (y) = x  F(x) = y, for all x  X and y  Y

39.  Cardinality is the number of things in a set  It is reflexive, symmetric, and transitive  Two sets have the same cardinality if a bijective function maps every element in one to an element in the other  Any set with the same cardinality as positive integers is called countably infinite

42.  Exam 2!

43.  Finish Homework 6  Due at Exam 2  Study for Exam 2  Friday in class