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CS322

Week 8 - Wednesday. CS322. 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!"

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CS322

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  1. Week 8 - Wednesday CS322

  2. Last time • What did we talk about last time? • Cardinality • Countability • Relations

  3. Questions?

  4. 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

  5. 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

  6. Reflexive, Symmetric, and Transitive Properties Student Lecture

  7. Properties of Relations

  8. 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 reflexiveiff 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

  9. Symmetric • Relation R is symmetriciff 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

  10. Transitive • Relation R is transitiveiff 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

  11. 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?

  12. 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

  13. 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

  14. Examples on real sets • Let R be a relation on real numbers R such that • xRy x = y • Is R reflexive? • Is R symmetric? • Is R transitive? • Let S be a relation on real numbers R such that • xSy x < y • Is S reflexive? • Is S symmetric? • Is S transitive? • Let T be a relation on positive integers N such that • mTn 3 | (m – n) • Is T reflexive? • Is T symmetric? • Is T transitive?

  15. Equivalence Relations

  16. 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

  17. 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

  18. 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)

  19. 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] = 

  20. Modular Arithmetic

  21. 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 equivalent • n | (a – b) • a b (mod n) • a = b + kn for some integer k • a and b have the same remainder when divided by n • a mod n = b mod n

  22. 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: • (a + b)  (c + d) (mod n) • (a – b)  (c – d) (mod n) • ab cd (mod n) • 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

  23. Partial Orders

  24. Antisymmetry • Let R be a relation on a set A • R is antisymmetriciff 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 Rantisymmetric? • Let S be the "divides" relation on the set of all integers • Is Santisymmetric?

  25. 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

  26. Quiz

  27. Upcoming

  28. Next time… • Review for Exam 2

  29. Reminders • Work on Homework 6 • Due Friday before midnight • Study for Exam 2 • Next Monday in class • Review on Friday

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