1. 2.1 Sets ‒Sets ‒Common Universal Sets ‒Subsets 2.2 Set Operations 2.3 Functions 2.4 Sequences and Summations 1

2. A set is a collection or group of objects or elements or members. (Cantor 1895) – A set is said to contain its elements. – There must be an underlying universal set U, either specifically stated or understood. Sets 2

3. Notation: – list the elements between braces: S = {a, b, c, d}={b, c, a, d, d} (Note: listing an object more than once does not change the set. Ordering means nothing.) – specification by predicates: S = {x| P(x)} S contains all the elements from U which make the predicate P true. – brace notation with ellipses: S = {..., -3, -2, -1}, the negative integers. 3

4. R = reals N = natural numbers = {0,1, 2, 3,... }, the counting numbers. Z = all integers = {.., -3, -2, -1, 0, 1, 2, 3,4,..}. Z + ={1,2,3,…},the set of positive integers. Q={P/q | p  Z,q  Z and q≠0},the set of rational numbers. Q + ={x  R | x=p/q, for some positive integers p and q} Common Universal Sets 4

5. Notation: x is a member of S or x is an element of S: x  S x is not an element of S: x  S Common Universal Sets 5

6. Definition: The set A is a subset of the set B, denoted A  B, iff  x[x  A→x  B] Definition: The void set, the null set, the empty set, denoted Ø, is the set with no members. Note: the assertion x  Ø is always false. Hence  x[x  Ø → x  B] is always true. Therefore, Ø is a subset of every set. Note: Set B is always a subset of itself. Subsets 6

7. Definition: If A  B but A  B the we say A is a proper subset of B, denoted A  B. Definition: The set of all subset of set A, denoted P(A), is called the power set of A. Subsets 7

8. Example: If A = {a, b} then P(A) = {Ø, {a}, {b}, {a,b}} Subsets 8

9. Definition: The number of (distinct) elements in A, denoted |A|, is called the cardinality of A. If the cardinality is a natural number (in N), then the set is called finite, else infinite. Subsets 9

10. Example: – A = {a, b}, |{a, b}| = 2, |P({a, b})| = 4. – A is finite and so is P(A). – Useful Fact: |A|=n implies |P(A)| = 2 n N is infinite since |N| is not a natural number. It is called a transfinite cardinal number. Note: Sets can be both members and subsets of other sets. Subsets 10

11. Example: A = {Ø,{Ø}}. A has two elements and hence four subsets: Ø, {Ø}, {{Ø}}, {Ø,{Ø}} Note that Ø is both a member of A and a subset of A! P. 1 Subsets 11

12. Example: A = {Ø}. Which one of the follow is incorrect: (a) Ø  A (b) Ø  A (c) {Ø}  A (d) {Ø}  A P. 1 Subsets 12

13. Russell's paradox: Let S be the set of all sets which are not members of themselves. Is S a member of itself? Another paradox: Henry is a barber who shaves all people who do not shave themselves. Does Henry shave himself? P. 1 Subsets 13

14. Definition: The Cartesian product of A with B, denoted A  B, is the set of ordered pairs { | a  A Λ b  B} Notation: A i ={ |a i  A i } Note: The Cartesian product of anything with Ø is Ø. (why?) P. 1 Subsets 14

15. Example: – A = {a,b}, B = {1, 2, 3} – AxB = {,,,,, } – What is BxA? AxBxA? If |A| = m and |B| = n, what is |AxB|? P. 1 Subsets 15

16. Sets Subset P. 1 Terms 16

17. 2.1 Sets 2.2 Set Operations –Set Operations –Venn Diagrams –Set Identities –Union and Intersection of Indexed Collections 2.3 Functions 2.4 Sequences and Summations P. 1 17

18. Propositional calculus and set theory are both instances of an algebraic system called a Boolean Algebra. The operators in set theory are defined in terms of the corresponding operator in propositional calculus. As always there must be a universe U. All sets are assumed to be subsets of U. P. 1 Set Operations 18

19. Definition: Two sets A and B are equal, denoted A = B, iff  x[x  A ↔ x  B]. Note: By a previous logical equivalence we have A = B iff  x[(x  A→ x  B) Λ (x  B → x  A)] or A = B iff A  B and B  A P. 1 Set Operations 19

20. Definitions: –The union of A and B, denoted A  B, is the set {x | x  A V x  B} –The intersection of A and B, denoted A  B, is the set {x | x  A Λ x  B} P. 1 Set Operations 20 FIGURE 1 FIGURE 2

21. Note: If the intersection is void, A and B are said to be disjoint. – The complement of A, denoted, is the set {x | x  A}={x | ¬(x  A)} Note: Alternative notation is, and {x|x  A}. – The difference of A and B, or the complement of B relative to A, denoted A - B, is the set A  Note: The complement of A is U - A. – The symmetric difference of A and B, denoted A  B, is the set (A - B)  (B - A). P. 1 Set Operations 21

22. Examples: U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} A= {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8}. Then – A  B = {1, 2, 3, 4, 5, 6, 7, 8} – A  B = {4, 5} – = {0, 6, 7, 8, 9, 10} – = {0, 1, 2, 3, 9, 10} – A - B = {1, 2, 3} – B - A = {6, 7, 8} – A  B = {1, 2, 3, 6, 7, 8} P. 1 Set Operations 22

23. A useful geometric visualization tool (for 3 or less sets) – The Universe U is the rectangular box. – Each set is represented by a circle and its interior. – All possible combinations of the sets must be represented. Shade the appropriate region to represent the given set operation. P. 1 Venn Diagrams 23

24. Set identities correspond to the logical equivalences. Example: The complement of the union is the intersection of the complements P. 1 Set Identities 24

25. Set Identities Proof: To show To show two sets are equal we show for all x that x is a member of one set if and only if it is a member of the other. We now apply an important rule of inference (defined later) called Universal Instantiation 25

26. Set Identities Universal Instantiation In a proof we can eliminate the universal quantifier which binds a variable if we do not assume anything about the variable other than it is an arbitrary member of the Universe. We can then treat the resulting predicate as a proposition. 26

27. We say 'Let x be arbitrary. ' Then we can treat the predicates as propositions: Assertion Reason Def. of complement ¬ Def. of   ¬ Def. of union  ¬ x  A Λ ¬ x  B DeMorgan's Laws Def. of  Def. of complement Def. of intersection Hence is a tautology. P. 1 Set Identities 27

28. Since – x was arbitrary – we have used only logically equivalent assertions and definitions we can apply another rule of inference called Universal Generalization We can apply a universal quantifier to bind a variable if we have shown the predicate to be true for all values of the variable in the Universe. and claim the assertion is true for all x, i.e., P. 1 Set Identities

29. Q. E. D. (an abbreviation for the Latin phrase “Quod Erat Demonstrandum” - “which was to be demonstrated” used to signal the end of a proof) Note: As an alternative which might be easier in some cases, use the identity 29

30. Example: Show A  (B - A) = Ø The void set is a subset of every set. Hence, A  (B - A)  Ø Therefore, it suffices to show A  (B - A)  Ø or  x[x  A  (B - A) → x  Ø] P. 1 Set Identities 30

31. Set Identities So as before we say 'let x be arbitrary’. Show x  A  (B- A) → x  Ø is a tautology. But the consequent is always false. Therefore, the antecedent better always be false also. 31

32. Solution: Apply the definitions: Assertion Reason x  A  (B- A)  x  A Λ x  (B- A) by Def. of   x  A Λ (x  B Λ x  A) by Def. of -  (x  A Λ x  A) Λ x  B by Def. of Associative  0 Λ x  B by Def. of Complement  0 (false) by Def. of Domination P. 1 Set Identities 32

33. Let A 1,A 2,..., A n be an indexed collection of sets. Definition: The union of a collection of sets is the set that contains those elements that are members of at least one set in the collection. we use the notation to denote the union of the sets A 1,A 2,..., A n P. 1 Union and Intersection of Indexed Collections 33

34. Union and Intersection of Indexed Collections Definition: The intersection of a collection of sets is the set that contains those elements that are members of all the set in the collection. we use the notation to denote the intersection of the sets A 1,A 2,..., A n 34

35. Examples: Let P. 1 Union and Intersection of Indexed Collections 35

36. Boolean Algebra Set operations Union Intersection Complement Difference P. 1 Terms 36 Symmetric difference Venn Diagram Set Identities Universal Instantiation Universal Generalization

37. 2.1 Sets 2.2 Set Operations 2.3 Functions ‒Functions ‒ Injections, Surjections and Bijections ‒ Inverse Functions ‒Composition 2.4 Sequences and Summations P. 1 37

38. Definition: Let A and B be sets. A function (mapping, map) f from A to B, denoted f :A→ B, is a subset of A×B such that and P. 1 Functions 38

39. Note: f associates with each x in A one and only one y in B. – A is called the domain – B is called the codomain. If f(x) = y – y is called the image of x under f – x is called a preimage of y (note there may be more than one preimage of y but there is only one image of x). P. 1 Functions 39

40. The range of f is the set of all images of points in A under f. We denote it by f(A). If S is a subset of A then f(S) = {f(s) | s in S}. Q: What is the difference between range and codomain? P. 1 Functions 40

41. Example: 1.f(a) = 2.the image of d is 3.the domain of f is 4.the codomain is 5.f(A) = 6.the preimage of Y is 7.the preimages of Z are 8.f({c,d}) = 41 P. 1 Functions Z Z A = {a, b, c, d} B = {X, Y, Z} {Y, Z} b a, c,d {Z}{Z}

42. Let f be a function from A to B. Definition: f is one-to-one (denoted 1-1) or injective if preimages are unique. Note: this means that if a  b then f(a)  f(b). Definition: f is onto or surjective if every y in B has a preimage. Note: this means that for every y in B there must be an x in A such that f(x) = y. Definition: f is bijective if it is surjective and injective (one-to-one and onto). P. 1 Injections, Surjections and Bijections 42

43. Examples: The previous Example function is neither an injection nor a surjection. Hence it is not a bijection. P. 1 Injections, Surjections and Bijections 43

44. Note: Whenever there is a bijection from A to B, the two sets must have the same number of elements or the same cardinality. That will become our definition, especially for infinite sets. P. 1 Injections, Surjections and Bijections 44

45. Examples: Let A = B = R, the reals. Determine which are injections, surjections, bijections: 1.f(x) = x, 2.f(x) = x 2, 3.f(x) = x 3, 4.f(x) = | x |, 5.f(x) = x + sin(x) P. 1 Injections, Surjections and Bijections 45

46. Let E be the set of even integers {0, 2, 4, 6,....}. Then there is a bijection f from N to E, the even nonnegative integers, defined by f(x) = 2x. Hence, the set of even integers has the same cardinality as the set of natural numbers. OH, NO! IT CAN’T BE....E IS ONLY HALF AS BIG!!! Sorry! It gets worse before it gets better. （見山是山，見水是水前，會先見山不是山，見 水不是水） P. 1 Injections, Surjections and Bijections 46

47. Definition: Let f be a bijection from A to B. Then the inverse of f, denoted f -1, is the function from B to A defined as f -1 (y) = x iff f(x) = y P. 1 Inverse Functions 47

48. Inverse Functions Example: Let f be defined by the diagram: Note: No inverse exists unless f is a bijection. 48 P. 1

49. Definition: Let S be a subset of B. Then f -1 (S) = {x | f(x)  S} Note: f need not be a bijection for this definition to hold. P. 1 Inverse Functions 49

50. Example: Let f be the following function: f -1 ({Z}) = f -1 ({X, Y}) = 50 P. 1 Inverse Functions {c, d} {a, b}

51. Definition: Let f: B→C, g: A → B. The composition of f with g, denoted f  g, is the function from A to C defined by f  g(x) = f(g(x)) P. 1 Composition 51

52. Examples: If f(x) = and g(x) = 2x + 1, then f(g(x)) = and g(f(x)) = 2 + 1 P. 1 Composition 52

53. Definition: The floor function, denoted f (x) =  x  or f(x) = floor(x), is the largest integer less than or equal to x. The ceiling function, denoted f (x) =  x  or f(x) = ceiling(x), is the smallest integer greater than or equal to x. Examples:  3.5  = 3,  3.5  = 4. Note: the floor function is equivalent to truncation for positive numbers. P. 1 53 Floor and Ceiling

54. Example: Suppose f: B→C, g: A→B and f  g is injective. What can we say about f and g? – We know that if a  b then f(g(a))  f(g(b)) since the composition is injective. – Since f is a function, it cannot be the case that g(a)= g(b) since then f would have two different images for the same point. – Hence, g(a)  g(b) – It follows that g must be an injection. – However, f need not be an injection (you show). P. 1 Composition 54

55. Function Image Preimage Range Injective (one-to- one) Surjective Bijective P. 1 Terms 55 Cardinality Inverse function Composition Floor function Ceiling function

56. 2.1 Sets 2.2 Set Operations 2.3 Functions 2.4 Sequences and Summations ‒ Sequences and Summations ‒ Summation Notation ‒ Cardinality ‒ Some Countably Infinite Sets ‒ Cantor Diagonalization P. 1

57. Definition: A sequence is a function from a subset of the natural numbers (usually of the form {0, 1, 2,... } to a set S. Note: the sets {0, 1, 2, 3,..., k} and {1, 2, 3, 4,..., k} are called initial segments of N. P. 1 Sequences and Summations

58. Notation: if f is a function from {0, 1, 2,...} to S we usually denote f(i) by a i and we write where k is the upper limit (usually ∞). P. 1 Sequences and Summations

59. Examples: Using zero-origin indexing, if f(i) = 1/(i + 1). then the sequence f = {1, 1/2, 1/3,1/4,... } = {a 0, a 1, a 2, a 3,.. } Using one-origin indexing the sequence f becomes {1/2, 1/3,...} = {a 1, a 2, a 3,...} P. 1 Sequences and Summations

60. Given a sequence we can add together a subset of the sequence by using the summation and function notation or more generally P. 1 Summation Notation

61. Examples: If S = {2, 5, 7, 10} then Similarly for the product notation: P. 1 Summation Notation

62. Definition: A geometric progression is a sequence of the form Your book has a proof that (you can figure out what it is if r = 1). You should also be able to determine the sum – if the index starts at k vs. 0 – if the index ends at something other than n (e.g., n-1, n+1, etc.). P. 1 Summation Notation

63. Definition: The cardinality of a set A is equal to the cardinality of a set B, denoted |A| = |B|, if there exists a bijection from A to B. Definition: If a set has the same cardinality as a subset of the natural numbers N, then the set is called countable. If |A| = |N|, the set A is countably infinite. The (transfinite) cardinal number of the set N is aleph null = ﭏ 0. If a set is not countable we say it is uncountable. P. 1 Cardinality

64. Examples: The following sets are uncountable (we show later) – The real numbers in [0, 1] – P(N), the power set of N Note: With infinite sets proper subsets can have the same cardinality. This cannot happen with finite sets. Countability carries with it the implication that there is a listing of the elements of the set. P. 1 Cardinality

65. Definition: | A | ≤ | B | if there is an injection from A to B. Note: as you would hope. Theorem: If | A | ≤ | B | and | B | ≤ | A | then | A | = | B |. This implies – if there is an injection from A to B – if there is an injection from B to A then – there must be a bijection from A to B This is difficult to prove but is an example of demonstrating existence without construction. It is often easier to build the injections and then conclude the bijection exists. P. 1 Cardinality

66. Example: Theorem: If A is a subset of B then | A | ≤ | B | Proof: the function f(x) = x is an injection from A to B. Example: | {0, 2, 5} | ≤ ﭏ 0 The injection f: {0, 2, 5} →N defined by f(x) = x is shown below: Cardinality

67. The set of even positive integers E ( 0 is considered even) is countably infinite. Note that E is a proper subset of N! Proof: Let f(x) = 2x. Then f is a bijection from N to E Z +, the set of positive integers is countably infinite. P. 1 Some Countably Infinite Sets

68. f(0) = 1/1, f(1) = 1/2, f(2) = 2/1, f(3) = 3/1, and so forth. Every rational number appears on the list at least once, some many times (repetitions). Hence, | N | = | Q R | = ﭏ 0. Q. E. D. P. 1 Some Countably Infinite Sets The position on the path (listing) indicates the image of the bijective function f from N to Q R : The set of positive rational numbers Q + is countably infinite. The set of all rational numbers Q, positive and negative, is countably infinite.

69. The set of (finite length) strings S over a finite alphabet A is countably infinite. To show this we assume that - A is nonvoid - There is an “alphabetical” ordering of the symbols in A Proof: List the strings in lexicographic order: – all the strings of zero length, – then all the strings of length 1 in alphabetical order, – then all the strings of length 2 in alphabetical order, etc. This implies a bijection from N to the list of strings and hence it is a countably infinite set. P. 1 Some Countably Infinite Sets

70. For example: Let A = {a, b, c}. Then the lexicographic ordering of A is {, a, b, c, aa, ab, ac, ba, bb, bc, ca, cb, cc, aaa, aab, aac, aba,....} ={f(0), f(1), f(2), f(3), f(4),....} P. 1 Some Countably Infinite Sets

71. The set of all C programs is countable. Proof: Let S be the set of legitimate characters which can appear in a C program. – A C compiler will determine if an input program is a syntactically correct C program (the program doesn't have to do anything useful). – Use the lexicographic ordering of S and feed the strings into the compiler. – If the compiler says YES, this is a syntactically correct C program, we add the program to the list. – Else we move on to the next string. In this way we construct a list or an implied bijection from N to the set of C programs. Hence, the set of C programs is countable. Q. E. D. P. 1 Some Countably Infinite Sets

72. An important technique used to construct an object which is not a member of a countable set of objects with (possibly) infinite descriptions. Theorem: The set of real numbers between 0 and 1 is uncountable. Proof: We assume that it is countable and derive a contradiction. If it is countable we can list them (i.e., there is a bijection from a subset of N to the set). We show that no matter what list you produce we can construct a real number between 0 and 1 which is not in the list. Hence, there cannot exist a list and therefore the set is not countable P. 1 Cantor Diagonalization

73. Represent each real number in the list using its decimal expansion. e.g., 1/3 =.3333333........ 1/2 =.5000000........ =.4999999........ If there is more than one expansion for a number, it doesn't matter as long as our construction takes this into account. THE LIST.... r1 =.d 11 d 12 d 13 d 14 d 15 d 16..... r2 =.d 21 d 22 d 23 d 24 d 25 d 26.... r3 =.d 31 d 32 d 33 d 34 d 35 d 36.... P. 1 Cantor Diagonalization It's actually much bigger than countable. It is said to have the cardinality of the continuum, c.

74. Now construct the number x =.x 1 x 2 x 3 x 4 x 5 x 6 x 7.... x i = 3 if d ii  3 x i = 4 if d ii = 3 Then x is not equal to any number in the list. Hence, no such list can exist and hence the interval (0,1) is uncountable. Q. E. D. P. 1 Cantor Diagonalization

75. Sequence Summation Cardinality Zero-origin indexing One-origin indexing Geometric progression Countable infinite Uncountable Cantor diagonalization P. 1 Terms Sequence Summation Cardinality Zero-origin indexing One-origin indexing Geometric progression Countable infinite Uncountable Cantor diagonalization