Lect1 No 873503264

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Lect1 No 873503264

  1. 1. A 2008 A
  2. 2. Georg Cantor: 1845-1918 A
  3. 3. Outline 1 2 3 A
  4. 4. a∈A a A A, B A = B iff (∀x)(x ∈ A ↔ x ∈ B).1 (1) ∅. 1 iff: if and only if A
  5. 5. {1, 2} 1 2 {1, 2} = {2, 1}. {1, 2, 2} = {1, 1, 1, 2} = {1, 2}. {dogs} {} A
  6. 6. ( ) {x : P(x)} ( {x|P(x)}) P {x : x } {x ∈ A : P(x)} A P x. {F (x) : x ∈ A} A F {2x : x ∈ Z} {F (x) : P(x)} P F {x 2 : x } A
  7. 7. ( ) A, B A B (subset) A B A B B A. A⊆B B ⊇ A. B B B A B A ⊂ B. A
  8. 8. A A ⊆ A, ∅ ⊆ A. R, Q, Z, N N⊂Z⊂Q⊂R A
  9. 9. U U U U A A( U ) A A = {x ∈ U : x ∈ A}, U A U {A : A ⊆ U}, U P(U) 2U . A
  10. 10. A, B A, B A B A ∪ B; A B A ∩ B; A B A − B. A ∪ B = {x : (x ∈ A) or (x ∈ B)}; A ∩ B = {x : (x ∈ A) and (x ∈ B)} = {x ∈ A : x ∈ B}; A − B = {x : (x ∈ A) and (x ∈ B)} = {x ∈ A : x ∈ B}. A
  11. 11. A∪∅=A A∩U =A A∪U =U A∩∅=∅ A∪A=A A∩A=A A∪B =B∪A A∩B =B∩A A∪A=U A∩A=∅ A
  12. 12. (A) = A A ∪ (A ∩ B) = A A ∩ (A ∪ B) = A A ∪ (B ∪ C) = (A ∪ B) ∪ C A ∩ (B ∩ C) = (A ∩ B) ∩ C A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) A∪B =A∩B A∩B =A∪B A
  13. 13. ( ) P(U) F (U ) F U A, B ∈ F A ∪ B, A ∈ F F0 = {∅, U} A
  14. 14. Outline 1 2 3 A
  15. 15. a, b (a, b). (a, b), (c, d) iff a = c, b = d A,B A × B = {(a, b) : (a ∈ A) and (b ∈ B)}. n A1 · · · An A1 × A2 × · · · × An n i=1 Ai . A
  16. 16. A, B, C A × ∅ = ∅ × A = ∅, A = ∅, B = ∅ A=B A × B = B × A. A × (B × C) = (A × B) × C; A × (B ∪ C) = (A × B) ∪ (A × C); A × (B ∩ C) = (A × B) ∩ (A × C); (B ∪ C) × A = (B × A) ∪ (C × A); (B ∩ C) × A = (B × A) ∩ (C × A). A
  17. 17. ( ) A, B A B R A×B A a B b a, b R (a, b) ∈ R. aRb R(a, b). A
  18. 18. A=B A × A, ∅, idA = {(a, a) : a ∈ A}. A B R R dom(R) = {x| y (x, y ) ∈ R}; ran(R) = {y | x (x, y ) ∈ R}. A
  19. 19. ( ) A B R B C S R R ∼ = {(x, y )|(y , x) ∈ R}. R S R ◦ S = {(x, z)| y (xRy ) (ySz)}. R∼ B A R◦S A C A
  20. 20. n- A
  21. 21. Example (Russell & Novig: AIMA, Chapter 5) Consider the following binary constraint problem P V = {WA, SA, NT , Q, NSW , V , T } U = {red, green, blue} C: no neighboring regions have the same color A
  22. 22. Outline 1 2 3 A
  23. 23. ( ) A B R A B x ∈A y ∈B (x, y ) ∈ R R dom(R) = A, x ∈ dom(R) y ∈ ran(R) (x, y ) ∈ R f , g, h A B f, f : A → B, (x, y ) ∈ f , f (x) = y . A
  24. 24. ( ) f :A→B b∈B a∈A b = f (a); f a, a ∈ A, f (a) = f (a ) a=a; f f A
  25. 25. f :A→B ran(f ) f ran(f ) = {b} ⊆ B f cb . A=∅ A B ∅; f A A a f (a) = a, f idA . A
  26. 26. n (n ) n ≥ 1, A n f : An → A n A 0 A A
  27. 27. ( ) f : A → B, g : B → C, g ◦ f : A → C (g ◦ f )(x) = g(f (x)).a a f g f ◦ g, g ◦ f. A
  28. 28. f,g g◦f g◦f g g◦f f g◦f f g A
  29. 29. U R (reflexive) x ∈U (x, x) ∈ R R idU ⊆ R. U R x, y ∈ U (x, y ) ∈ R (y , x) ∈ R R = R∼. U R x, y ∈ U, (x, y ) ∈ R (y , x) ∈ R x =y R ∩ R ∼ ⊆ idU . U R x, y , z ∈ U, (x, y ) ∈ R (y , z) ∈ R (x, z) ∈ R, R ◦ R ⊆ R. A
  30. 30. ( ) U R R R U A
  31. 31. R U x ∈U [x]R x [x]R = {y ∈ U : xRy }. U R ( ) U R U/R = {[x]R : x ∈ U}. U R A
  32. 32. ( ) π U π ⊆ P(U) π U π U, π π U A
  33. 33. U π U Rπ U a, b Rπ iff a, b π U R, U/R U πR ; πR R. A
  34. 34. ( ) A
  35. 35. ( ) P U, U P U R R P- P R U r (R) R r (R) = R ∪ idA . s(R) R s(R) = R ∪ R ∼ . t(R) R ∞ t(R) = R ∪ R 2 ∪ R 3 ∪ · · · = Ri . i=1 A
  36. 36. X X , a a; if a b and b a then a = b; if a b and b c then a c. X (partially ordered set, or poset) A
  37. 37. An example of poset The Hasse diagram of (℘({x, y , z}), ⊆)2 2 http://en.wikipedia.org/wiki/Hasse_diagram A
  38. 38. Total order and well-order A partial order is total (or linear) if for any a, b ∈ X , a b or b a is a well-order if every nonempty subset Y of X has a least element A
  39. 39. Tree A (rooted) tree is a poset (T , ) such that T has a unique least element, called the root the predecessors of every node are well ordered by A path on a tree T is a maximally linearly ordered subset of T . A
  40. 40. Group A group is a nonempty set G with a binary operation ◦ : G × G → G such that (a ◦ b) ◦ c = a ◦ (b ◦ c) for all a, b, c ∈ G. An element e in G is called an identity if e ◦ x = x ◦ e for any x. A semi-group that has an identity is called a monoid. A semi-group with an identity e is a group if each element x has a unique inverse y such that x ◦ y = y ◦ x = e. A

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