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교환 학생 프로그램

교환 학생 프로그램. 내년 1월 중순부터 6월 초 현재 학부 2,3 학년? Department of Information Technology, Uppsala University, Sweden (http://www.it.uu.se?lang=en) 마감이 11월 30일 민상렬 교수님 symin@snu.ac.kr 011-9120-7047, 또는 감지혜 조교 (880-7287,7288). 교환 학생 프로그램. 정통부의 지원 본인 부담 비용: 약 500만원 120만원 : 비행기표

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교환 학생 프로그램

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  1. 교환 학생 프로그램 • 내년 1월 중순부터 6월 초 • 현재 학부 2,3 학년? • Department of Information Technology, Uppsala University, Sweden (http://www.it.uu.se?lang=en) • 마감이 11월 30일 • 민상렬 교수님 • symin@snu.ac.kr • 011-9120-7047, • 또는 감지혜 조교 (880-7287,7288)

  2. 교환 학생 프로그램 • 정통부의 지원 • 본인 부담 비용: 약 500만원 • 120만원: 비행기표 • 6개월 전체 생활비 약 800만원 중 420만원은 정부 보조, 나머지 380만원 본인 부담, 등록금은 없음 • 서울대학교에서도 등록을 하지 않아도 되므로, 그리고 실제로 국내에 있어도 생활비와 용돈은 필요하므로 개인 부담은 전혀 없는 것과 마찬가지임

  3. 지원 자격 • IT전공 학생으로서 직전 학기까지 평균성적이 80% 이상인 자: 서울대학교 GPA 2.8~2.9정도에 해당 • TOEFL CBT 213점(PBT 550점), TOEIC 750점, TEPS 656점, G-TELP 92점(Level 3), 69점 (Level 2) 이상인 자 ※ 공인 어학성적은 신청마감일(11월 30일) 을 기준으로 최근 2년 이내의 성적을 인정

  4. Module #18:Relations- part II Rosen 5th ed., ch. 7

  5. §7.4: Closures of Relations • For any property X, the “X closure” of a set A is defined as the “smallest” superset of A that has the given property. • The reflexive closure of a relation R on A is obtained by adding (a,a) to R for each aA. I.e.,it is R  IA • The symmetric closure of R is obtained by adding (b,a) to R for each (a,b) in R. I.e., it is R  R−1 • The transitive closure or connectivity relation of R is obtained by repeatedly adding (a,c) to R for each (a,b),(b,c) in R. • I.e., it is

  6. Paths in Digraphs/Binary Relations • A path of length n from node a to b in the directed graph G (or the binary relation R) is a sequence (a,x1), (x1,x2), …, (xn−1,b) of n ordered pairs in EG (or R). • An empty setof edges is considered a path of length 0 from a to a. • If any path from a to b exists, then we say that a is connected tob. (“You can get there from here.”) • A path of length n≥1 from a to a is called a circuit or a cycle. • Note that there exists a path of length n from a to b in R if and only if (a,b)Rn.

  7. Simple Transitive Closure Alg. A procedure to compute R* of n elements with 0-1 matrices. proceduretransClosure(MR:rank-n 0-1 mat.) A := B := MR; fori := 2 to nbeginA := A⊙MR; B := B A {join}endreturn B {Alg. takes Θ(n4) time} {note: A represents Ri}

  8. Warshall’s Algorithm • Uses only Θ(n3) operations! Procedure Warshall(MR : rank-n 0-1 matrix) W := MR fork := 1 tonfori := 1 tonforj := 1 tonwij := wij (wik  wkj)return W {this represents R*} wij = 1 means there is a path from i to j going only through nodes ≤k

  9. §7.5: Equivalence Relations • An equivalence relation (e.r.) on a set A is simply any binary relation on A that is reflexive, symmetric, and transitive. • E.g., = itself is an equivalence relation. • For any function f:A→B, the relation “have the same f value”, or = {(a1,a2) | f(a1)=f(a2)} is an equivalence relation, e.g., let m=“mother of” then = “have the same mother” is an e.r.

  10. Equivalence Relation Examples • “Strings a and b are the same length.” • “Integers a and b have the same absolute value.” • “Real numbers a and b have the same fractional part (i.e., a − b Z).” • “Integers a and b have the same residue modulo m.” (for a given m>1)

  11. Equivalence Classes • Let R be any equiv. rel. on a set A. • The equivalence class of a, [a]R :≡ { b | aRb } (optional subscript R) • It is the set of all elements of A that are “equivalent” to a according to the eq. rel. R. • Each such b (including a itself) is called a representative of [a]R. • Since f(a)=[a]Ris a function of a, any equivalence relation R be defined using aRb :≡ “a and b have the same f value”, given that f. A set is divided into one or more classes

  12. Equivalence Class Examples • “Strings a and b are the same length.” • [a] = the set of all strings of the same length as a. • “Integers a and b have the same absolute value.” • [a] = the set {a, −a} • “Real numbers a and b have the same fractional part (i.e., a − b Z).” • [a] = the set {…, a−2, a−1, a, a+1, a+2, …} • “Integers a and b have the same residue modulo m.” (for a given m>1) • [a] = the set {…, a−2m, a−m, a, a+m, a+2m, …}

  13. Equivalent classes • Let R be an equivalence relation on the set A, then the following statements are equivalent • aRb • [a] = [b] • [a]  [b]  

  14. Partitions • A partition of a set A is the set of all the equivalence classes {A1, A2, … } for some e.r. on A. • The Ai’s are all disjoint and their union = A. • They “partition” the set into pieces. Within each piece, all members of the set are equivalent to each other.

  15. §7.6: Partial Orderings (POSET) A relation R on a set S is called a partial ordering or partial order if it is reflexive, antisymmetric, and transitive. A set S together with a partial ordering R is called a partially ordered set, or poset, and is denoted by (S, R)

  16. 1 2 3 Example Let S = {1, 2, 3} and let R = {(1,1), (2,2), (3,3), (1, 2), (3,1), (3,2) }

  17. In a poset, the notation ab denotes that This notation is used because the “less than or equal to” relation is a paradigm for a partial ordering. (Note that the symbol is used to denote the relation in any poset, not just the “less than or equals” relation.) The notation ab denotes thatab, but

  18. 2 2 3 2 1 2 3 Example Let S = {1, 2, 3} and let R = {(1,1), (2,2), (3,3), (1, 2), (3,1), (3,2)}

  19. Comparable/Incomparable The elements a and b of a poset (S, ) are called comparable if either ab or ba. When a and b are elements of S such that neither ab nor ba, a and b are called incomparable.

  20. Consider the power set of {a, b, c} and the subset relation. (P({a,b,c}), ) Example So, {a,c} and {a,b} are incomparable

  21. (S, ) Totally Ordered, Chains If is a poset and every two elements of S are comparable, S is called totally ordered or linearly ordered set, and is called a total order or a linear order. A totally ordered set is also called a chain. If (A,R) is a poset, we say that A is totally ordered if for all x,y in A either xRy or yRx. In this case R is called a total order.

  22. (S, ) Well-Ordered Set is a well-ordered set if it is a poset such that is a total ordering and such that every nonempty subset of S has a least element. 1 1 2 2 3 3

  23. Hasse Diagrams Given any partial order relation defined on a finite set, it is possible to draw the directed graph so that all of these properties are satisfied. This makes it possible to associate a somewhat simpler graph, called a Hasse diagram, with a partial order relation defined on a finite set.

  24. Hasse Diagrams (continued) • Start with a directed graph of the relation in which all arrows point upward. Then eliminate • the loops at all the vertices, • all arrows whose existence is implied by the transitive property, • the direction indicators on the arrows.

  25. Example Let A = {1, 2, 3, 9, 18} and consider the “divides” relation on A: For all 18 If aRb, normally a is located lower than b 9 2 3 1

  26. Example Eliminate the loops at all the vertices. Eliminate all arrows whose existence is implied by the transitive property. Eliminate the direction indicators on the arrows. 18 18 9 9 2 2 3 3 1 1

  27. Maximal and Minimal Elements a is a maximal in the poset (S, ) if there is nosuch that a b. Similarly, an element of a poset is called minimal if it is not greater than any element of the poset. That is, a is minimal if there is no element such that b a. It is possible to have multiple minimals and maximals.

  28. Greatest Element, Least Element a is the greatest element in the poset (S, ) if b a for all . Similarly, an element of a poset is called the least element if it is less than all other elements in the poset. That is, a is the least element if a b for all v e f u d w Hasse diagrams c y z a b x

  29. Upper bound, Lower bound Sometimes it is possible to find an element that is greater than all the elements in a subsetAof a poset (S, ). Ifu is an element of S such that au for all elements , then u is called an upper boundof A. Likewise, there may be an element less than all the elements in A. If l is an element of S such that la for all elements , then l is called a lower bound of A.

  30. The element x is called the least upper bound of the subset A if x is an upper bound that is less than every other upper bound of A. The element y is called the greatest lower bound of A if y is a lower bound of A and z y whenever z is a lower bound of A. Least Upper Bound,Greatest Lower Bound

  31. Find the glb and lub of {y,m,u}, if they exist v p The upper bounds are u,v. Since u  v, u is the lub u w m n The lower bounds are x,y. Since x  y, y is the glb y z x How about glb and lub of {m,n,w}?

  32. Lattices A partially ordered set in which every pair of elements has both a least upper bound and a greatest lower bound is called a lattice.

  33. Are they lattices? v v p v z u u u w w w m n m n n m y y z z x x x

  34. schedule • 11/25 graph-part I • 11/30 no class due to SNU entrance exam • 12/2 graph-part II, tree • 12/7 final exam

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