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04 - Sets

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I used this set of slides for the lecture on Sets I gave at the University of Zurich for the 1st year students following the course of Formale Grundlagen der Informatik.

Publié dans : Technologie
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04 - Sets

2. 2. What exactly is logic?
3. 3. What exactly is logic? the study of the principles of correct reasoning
4. 4. Wax on … wax off … these are the basics http://www.youtube.com/watch?v=3PycZtfns_U
5. 5. Sets www.tudorgirba.com
6. 6. computer information information computation
7. 7. Set A set is a group of objects.
8. 8. Set A set is a group of objects. {10, 23, 32}
9. 9. Set A set is a group of objects. {10, 23, 32} N = {0, 1, 2, … }
10. 10. Set A set is a group of objects. {10, 23, 32} N = {0, 1, 2, … } Z = {… , -2, -1, 0, 1, 2, … }
11. 11. Set A set is a group of objects. {10, 23, 32} N = {0, 1, 2, … } Z = {… , -2, -1, 0, 1, 2, … } Ø empty set
12. 12. Set A set is a group of objects. {10, 23, 32} N = {0, 1, 2, … } Z = {… , -2, -1, 0, 1, 2, … } Ø U empty set universe
13. 13. Set A set is a group of objects. {10, 23, 32} N = {0, 1, 2, … } Z = {… , -2, -1, 0, 1, 2, … } Ø U empty set universe Membership a is a member of set A
14. 14. Set A set is a group of objects. {10, 23, 32} N = {0, 1, 2, … } Z = {… , -2, -1, 0, 1, 2, … } 10 ∈ {10, 23, 32} Ø U empty set universe Membership a is a member of set A
15. 15. Set A set is a group of objects. {10, 23, 32} N = {0, 1, 2, … } Z = {… , -2, -1, 0, 1, 2, … } 10 ∈ {10, 23, 32} -1 ∉ N Ø U empty set universe Membership a is a member of set A
16. 16. Subset A⊆B Every member of A is also an element of B.
17. 17. Subset A⊆B ∀x:: x∈A x∈B Every member of A is also an element of B.
18. 18. Subset A⊆B ∀x:: x∈A x∈B ∅ ⊆ A. A ⊆ A. A = B A ⊆ B ∧ B ⊆ A. Every member of A is also an element of B.
19. 19. Subset A⊆B ∀x:: x∈A x∈B ∅ ⊆ A. A ⊆ A. A = B A ⊆ B ∧ B ⊆ A. Proper subset A⊂B A is a subset of B and not equal to B. Every member of A is also an element of B.
20. 20. Subset A⊆B ∀x:: x∈A x∈B ∅ ⊆ A. A ⊆ A. A = B A ⊆ B ∧ B ⊆ A. Proper subset A⊂B ∀x:: A⊆B ∧ A≠B A is a subset of B and not equal to B. Every member of A is also an element of B.
21. 21. Union A∪B ∀x:: x∈A ∨ x∈B A∪B={ x | x∈A or x∈B }
22. 22. Union A∪B ∀x:: x∈A ∨ x∈B A∪B={ x | x∈A or x∈B }
23. 23. Union A∪B ∀x:: x∈A ∨ x∈B A∪B={ x | x∈A or x∈B } A ∪ B = B ∪ A. A ∪ (B ∪ C) = (A ∪ B) ∪ C. A ⊆ (A ∪ B). A ∪ A = A. A ∪ ∅ = A. A ⊆ B A ∪ B = B.
24. 24. Intersection A∩B ∀x:: x∈A ∧ x∈B A∩B={ x | x∈A and x∈B }
25. 25. Intersection A∩B ∀x:: x∈A ∧ x∈B A∩B={ x | x∈A and x∈B }
26. 26. Intersection A∩B ∀x:: x∈A ∧ x∈B A∩B={ x | x∈A and x∈B } A ∩ B = B ∩ A. A ∩ (B ∩ C) = (A ∩ B) ∩ C. A ∩ B ⊆ A. A ∩ A = A. A ∩ ∅ = ∅. A ⊆ B A ∩ B = A.
27. 27. Complements AB, A’ ∀x:: x∈A ∧ x∉B AB={ x | x∈A and x∉B }
28. 28. Complements AB, A’ ∀x:: x∈A ∧ x∉B AB={ x | x∈A and x∉B } A B ≠ B A. A ∪ A′ = U. A ∩ A′ = ∅. (A′)′ = A. A A = ∅. U′ = ∅. ∅′ = U. A B = A ∩ B′.
29. 29. A ∩ U = A A ∪ ∅ = A Neutral elements
30. 30. A ∩ U = A A ∪ ∅ = A Neutral elements A ∩ ∅ = ∅ A ∪ U = U Zero elements
31. 31. A ∩ U = A A ∪ ∅ = A Neutral elements A ∩ ∅ = ∅ A ∪ U = U Zero elements A ∩ A = A A ∪ A = A Idempotence
32. 32. A ∩ U = A A ∪ ∅ = A Neutral elements A ∩ ∅ = ∅ A ∪ U = U Zero elements A ∩ A = A A ∪ A = A Idempotence A ∪ B = B ∪ A A ∩ B = B ∩ A Commutativity
33. 33. A ∩ U = A A ∪ ∅ = A Neutral elements A ∩ ∅ = ∅ A ∪ U = U Zero elements A ∩ A = A A ∪ A = A Idempotence A ∪ B = B ∪ A A ∩ B = B ∩ A Commutativity A ∩ (B ∩ C) = (A ∩ B) ∩ C A ∪ (B ∪ C) = (A ∪ B) ∪ C Associativity
34. 34. A ∩ U = A A ∪ ∅ = A Neutral elements A ∩ ∅ = ∅ A ∪ U = U Zero elements A ∩ A = A A ∪ A = A Idempotence A ∪ B = B ∪ A A ∩ B = B ∩ A Commutativity A ∩ (B ∩ C) = (A ∩ B) ∩ C A ∪ (B ∪ C) = (A ∪ B) ∪ C Associativity A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) Distributivity
35. 35. A ∩ U = A A ∪ ∅ = A Neutral elements A ∩ ∅ = ∅ A ∪ U = U Zero elements A ∩ A = A A ∪ A = A Idempotence A ∩ A’ = ∅ A ∪ A’ = U Complement A ∪ B = B ∪ A A ∩ B = B ∩ A Commutativity A ∩ (B ∩ C) = (A ∩ B) ∩ C A ∪ (B ∪ C) = (A ∪ B) ∪ C Associativity A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) Distributivity
36. 36. Similar to boolean algebra a ∧ 1 = a a ∨ 0 = a Neutral elements a ∧ 0 = 0 a ∨ 1 = 1 Zero elements a ∧ a = a a ∨ a = a Idempotence a ∧ ¬ a = 0 a ∨ ¬ a = 1 Negation a ∨ b = b ∨ a a ∧ b = b ∧ a Commutativity a ∧ (b ∧ c) = (a ∧ b) ∧ c a ∨ (b ∨ c) = (a ∨ b) ∨ c Associativity a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c) a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c) Distributivity
37. 37. A ∩ U = A A ∪ ∅ = A Neutral elements A ∩ ∅ = ∅ A ∪ U = U Zero elements A ∩ A = A A ∪ A = A Idempotence A ∩ A’ = ∅ A ∪ A’ = U Complement A ∪ B = B ∪ A A ∩ B = B ∩ A Commutativity A ∩ (B ∩ C) = (A ∩ B) ∩ C A ∪ (B ∪ C) = (A ∪ B) ∪ C Associativity A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) Distributivity
38. 38. A ∩ U = A A ∪ B = B ∪ A A ∪ ∅ = A A ∩ ∅ = ∅ A ∪ U = U A ∩ A = A A ∪ A = A A ∩ A’ = ∅ A ∪ A’ = U Neutral elements Zero elements Idempotence Complement A ∩ (B ∩ C) = (A ∩ B) ∩ C A ∩ B = B ∩ A 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’) Commutativity Associativity Distributivity DeMorgan’s
39. 39. A ⊆ A. A ⊆ B ∧ B ⊆ A A = B. A ⊆ B ∧ B ⊆ C A ⊆ C Reﬂexivity Anti-symmetry Transitivity
40. 40. Scissors Paper Stone
41. 41. Scissors Paper Stone beats beats beats
42. 42. Scissors Paper Stone beats beats beats
43. 43. Scissors Paper Stone beats beats beats beats Scissors Paper Stone Scissors FALSE TRUE FALSE Paper FALSE FALSE TRUE Stone TRUE FALSE FALSE
44. 44. Scissors Paper Stone beats beats beats beats Scissors Paper Stone Scissors FALSE TRUE FALSE Paper FALSE FALSE TRUE Stone TRUE FALSE FALSE
45. 45. Scissors Paper Stone beats beats beats beats Scissors Paper Stone Scissors FALSE TRUE FALSE Paper FALSE FALSE TRUE Stone TRUE FALSE FALSE
46. 46. Scissors Paper Stone beats beats beats beats Scissors Paper Stone Scissors FALSE TRUE FALSE Paper FALSE FALSE TRUE Stone TRUE FALSE FALSE
47. 47. beats Scissors Paper Stone Scissors FALSE TRUE FALSE Paper FALSE FALSE TRUE Stone TRUE FALSE FALSE beats = {(Scissors, Paper), (Paper, Stone), (Stone, Scissors)}
48. 48. beats Scissors Paper Stone Scissors FALSE TRUE FALSE Paper FALSE FALSE TRUE Stone TRUE FALSE FALSE beats = {(Scissors, Paper), (Paper, Stone), (Stone, Scissors)} beats ⊆ {Scissor, Paper, Stone} x {Scissor, Paper, Stone}
49. 49. Cartesian product AxB AxB={ (a,b) | a∈A and b∈B }
50. 50. Cartesian product AxB AxB={ (a,b) | a∈A and b∈B } A × ∅ = ∅. A × (B ∪ C) = (A × B) ∪ (A × C). (A ∪ B) × C = (A × C) ∪ (B × C).
51. 51. N-ary Relation A1, A2, ..., An R ⊆ A1 x A2 x...x An
52. 52. N-ary Relation A1, A2, ..., An R ⊆ A1 x A2 x...x An Binary Relation
53. 53. N-ary Relation A1, A2, ..., An R ⊆ A1 x A2 x...x An Binary Relation A1, A2 R ⊆ A1 x A2 (a,b) ∈ R aRb
54. 54. N-ary Relation A1, A2, ..., An R ⊆ A1 x A2 x...x An Binary Relation A1, A2 R ⊆ A1 x A2 (a,b) ∈ R aRb
55. 55. Tudor Gîrba www.tudorgirba.com creativecommons.org/licenses/by/3.0/