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Chapter 2 Basic Structures: Sets, Functions, Sequences and Sums (and part of Chapter 3)

Chapter 2 Basic Structures: Sets, Functions, Sequences and Sums (and part of Chapter 3). By courtesy of Prof. Cheng-Chia Chen. 2.1 Sets. Basic structure upon which all other (discrete and continuous) structures are built. A set is a collection of objects.

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Chapter 2 Basic Structures: Sets, Functions, Sequences and Sums (and part of Chapter 3)

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  1. Chapter 2 Basic Structures: Sets, Functions, Sequences and Sums(and part of Chapter 3) By courtesy of Prof. Cheng-Chia Chen

  2. 2.1 Sets • Basic structure upon which all other (discrete and continuous) structures are built. • A set is a collection of objects. • An object is anything of interest, maybe itself a set. • Definition 1. • A set is a collection of objects. • The objects in a set are called the elements or members of the set. • If x is a member of a set S, we say S contains x. • Notation: x Î S vs x Ï S • Ex: In 1, 2, 3, 4, 5, the collection of 1, 3 and 5 is a set.

  3. Set description • How to describe a set? 1. List all its member(s). • The set of all positive odd integer < 10 =? • The set all decimal digits =? • The set of all upper case English letters =? • The set of all nonnegative integers =? 2. Set builder notation: • P(x): a property (or a statement or a proposition) about objects, e.g., P(x) = “x > 0 and x is odd” • Then {x | P(x)} is the set of objects satisfying property P. • P(3) is true => 3 Î {x | P(x)} • P(2) is false => 2 Ï {x | P(x)}

  4. Conventions • N =def {x | x is a natural numbers} = {0,1,2,3,...} • N+ =def {1,2,3,...} • Z =def {...,-3,-2,-1,0,1,2,3,...} • Q =def the set of rational numbers • R =def the set of real numbers. • Problem: The same set may have many different descriptions. • {x | 0 < x <10 /\ x is odd} • {1,3,5,7,9}, {5,3,1,9,7} • {9,7,1,3,5}.

  5. Set predicates Definition 2. • Two sets S1, S2 are equal iff they have the same elements. • S1 = S2 iff "x (x Î S1 <-> x Î S2). • Ex: {1,3,5} = {1,5,3} = {1,1,3,3,5} • Graphical representation of sets: • Venn Diagrams: • rectangle: universal set U • circles: sets • points: particular elements • point x inside circle S => x Î S • point x outside S => x Ï S. U x S y S2 z w

  6. Set predicates (cont’d) • Null set = {} = Æ =def the collection of no objects. Def 3’ [empty set]: for-all x, x Ï Æ. Def 3 [subset]: • A Í B iff all elements of A are elements of B. • A Í B  for-all x (x Î A ⇒ x Î B). Def 3’’ [proper subset]: A Ì B =def A Í B /\ A ¹ B. • Exercise: Show that • 1. For all set A (Æ Í A): Pf: Let x be any element => x Ï Æ (by def) => x ÎÆ ⇒ x Î A holds. Hence x (x Î Æ ⇒ x Î A) and Æ Í A. • 2. (A Í B /\ B Í A)  (A = B) • 3. A Í Æ ⇒A = Æ (because Æ Í A) • Diagram representation of set inclusion relationship

  7. Size or cardinality of a set Def. 4: |A| = the size (cardinality) of A = # of distinct elements of A. • Ex: • |{1,3,3,5}| =? |{}| =? • |the set of binary digits| =? • |N| =? ; |Z| =? ; |{2i | i ∈ N}| =? • |R| =? Def. 5. • A set A is finite iff |A| is a natural number; otherwise it is infinite. • Two sets are of the same size (cardinality) iff there is a 1-1 & onto mapping between them.

  8. Countability of sets Def. • A set A is said to be denumerable (or countably infinite) iff |A| = |N|. • A set is countable iff either |A| = n for some n in N or |A| = |N|. • Exercises: Show that • 1. |N| = |Z| = |Q| = |{x | x  N and x >3 }| • 2. |R| = |(-1,1)| = |(0,1)| (define a 1-1 & onto mapping ) • 3. |(0,1)| is uncountable (by diagonalization) By exercises 1, 2 and 3 • R is not countable. • Q and Z are countable.

  9. The power set Def 6. • If A is a set, then the collection of all subsets of A is also a set, called the power set of A and is denoted as P(A) or 2A. • Ex: • P({0,1,2}) =? • P({}) =? • |P({1,2,...,n})| =? • Elements in a set are not ordered. But sometimes we need to distinguish between (1,3,4) and (3,4,1) --> ordered n-tuples

  10. Cartesian Products Def. 7 [n-tuple] • If a1,a2,...,an (n > 0) are n objects, then “(a1,a2,...,an)” is a new object, called an (ordered) n-tuple [with ai as its ith element] • Any ordered 2-tuple is called a pair. • (a1,a2,...,am) = (b1,b2,...,bn) means • (1) m = n and • (2) ai = bi for all 1 ≤ i ≤ m.

  11. Cartesian product Def. 8: [Cartesian product] A x B =def {(a,b) | a ∈ A and b ∈ B} A1 x A2 x ... x An =def {(a1,...,an) | ai ∈ Ai for all 1 ≤ i ≤ n}. Ex: A = {1,2}, B = {a,b,c} , C = {0,1} 1. A x B =? ; 2. B x A =? 3. A x {} =? ({}); 4. A x B x C =? Def. 8.1: Any subset of A x B is called a relation from A to B. (skip) Problems: 1. When will A x B = B x A? (A = B or one of them is {}.) 2. |A x B| =?

  12. The diagonalization principle (D.P.) (skip) • R: a binary relation on A (i.e. a subset of AxA) . D (the diagonal set for R ) =def { x | x Î A and (x,x) Ï R}. For each x Î A, let Ra (the raw of a) = { b Î A | (a,b) Î R}. Then D ≠ Ra for all a Î A. Example: Let A = {a,b,c,d,e,f} and R = Ra ={b,d} Rb={b,c} Rc={c} Rd={b,c,e,f} Re={e,f} Rf={a,c,d,e} D = {a,d,f}. D¹Rx since for each xÎA, xÎD iff (x,x)ÏR iff xÏRx.

  13. Generalization of the diagonalization principle (skip) A, B : two sets with a mapping f: A B. R :a relation from B to A (i.e., a subset of BxA). For all x ∈ A, let • Rf(x) ≡ {y | y ∈ A ∧ (f(x),y) ∈ R} ⊆ A, and • D ≡ {x | x ∈ A ∧ (f(x),x) ∉ R} ⊆ A. Then • D ≠ Rf(x) for all x ∈ A. Pf: Analogous to the previous one. Notes: 1. C = {Rf(x) | x ∈ A} ⊆ 2A is a set of subsets of A. 2. C ≠ 2A (∵ D ∈ 2A but D ∉ C).

  14. Generalization of the diagonalization principle (skip) f:A B ; R: BxA Rfa ={b,d} Rfb={b,c} Rfc={c} Rfd={b,c,e,f} Rfe={e,f} Rff={a,c,d,e} D = {a,d,f}. D¹Rfx since for each f(x)ÎB, xÎD iff (f(x),x)ÏR iff xÏRf(x).

  15. Show that |A| ≠ |2A| (skip) Pf: (1) The case that A is finite is trivial since |2A| = 2|A| > |A| and there is no bijection b/t two finite sets with different sizes. (2) Assume |A| = |2A|, i.e., there is a bijection f: A B, where B is 2A and R = { (y, x) | yB and x A and x y } Define the so-called diagonal setD in terms of f as follows: D = {x ∈ A | x Ï Rf(x) = f(x) }. Now (*) D is a subset of A and (**) D f(x) for any x ∈ A: since x ∈ D iff x Ï f(x) for each x ∈ A. ==> D is not a subset of A (∵ f is onto to 2A), a contradiction to (*). Hence f must not exist!!

  16. Application of the diagonalization principle (skip) • Theorem: The set 2N is uncountable. • pf: 1. direct from |A| ¹ |2A| with A = N. • 2. another proof: suppose 2N is denumerable. Then • there is a bijection f: N -> 2N. • let 2N = {S0,S1,S2,...} where Si = f(i). • Now the diagonal set • = { k | k Ï Sk }. By diagonalization principle, D¹ Sk for any k, but since D is a subset of N, by assumption, • D must equal to Sk for • some k. a contradiction!

  17. Application of the diagonalization principle (skip) Theorem: The real set (0,1) is uncountable. pf: If n is a real in (0,1), then it can be represented as an infinite sequence n = 0.n1n2n3… where each nk is one of {0,…,9}. Ex: 0 = 0.00000 ; 1 = 0.9999… ½ = 0.500000… = 0.49999…. 1/3 = 0.3333… p-3 = 0.14159…. For the representation to be unique, we exclude any sequence of the form: 0.d1d2…dk9999….and say any sequence not in such form a normal sequence. Now suppose the set (0,1) is denumerable. Then there must exist a bijection f: N  (0,1).

  18. (skip) • We apply diagonalization principle to define a number d from f as follows: let d = 0.d0d1d2…. where dk = (s == 0)? 8 : 9 – s, where s=f(k)k is the kth digit of the fraction part of f(k)

  19. (skip) • Now it can be shown that • d is a normal sequence representing a real number in (0,1) since it contains no 9. • d ¹ f(n) for all n since by construction dn¹ f(n)n, where f(n)k is the kth digit of the fraction part of f(n). This contradicts the facts that d ∈ (0,1) and f(N) = (0,1).

  20. Program that you are asked to design P HALT(P,X) P(x:String) { ….. } P(X) Halt? true yes false no “It’s a test …” X The Halting Problem (Section 3.1, 6th edition; p. 176) (Chapter 3) • L: any of your favorite programming languages (C, C++, Java, BASIC, etc. ) • Problem: write an L-program HALT(P,X), which takes another L-program P(-) and string X as input, and HALT(P,X) returns true if P(X) halts and returns false if P(X) does not halt.

  21. How hard is it to predict if a program will halt? (Chapter 3) • Consider the program int f( int n ) { if ( n == 1 ) return 1 ; if (n % 2 == 0 ) { // n is even  n/2 f(n % 2) ; } else { // n is odd  3n+1 f( n x 3 + 1) ; }} • It is conjectured that f(n) halts (returns 1) for all n ≥ 1. Ex: f(10)  5  16  8  4  2  1 (halts)

  22. Halt(P,X) does not exist (Chapter) • Ideas leading to the proof: Problem1: What about the truth value of the sentence: L: L is false Problem 2: Let S = {X | X X}. Then does S belong to S or not? The analysis: S  S => S  S; S  S => S  S. Problem 3: 矛盾說:1. 我的矛無盾不穿　2. 我的盾可抵擋所有茅 　　　結論: 1. 2. 不可同時為真。 Problem 4: 萬能上帝: 萬能上帝無所不能 => 可創造一個不服從他的子民 => 萬能上帝無法使所有子民服從 => 萬能上帝不是萬能 . 　　結論: 萬能上帝不存在。 Conclusion: • 1. S is not a set!! • 2. If a language is too powerful, it may produce expressions that is meaningless or can not be realized. • Question: If HALT(P,X) can be programmed, will it incur any absurd result like the case of S? Ans: yes!!

  23. H(P) HALT(P,X) loop P P(P) Halt? true yes P(x:String) { ….. } P halt false no P H(P): a program that HALT(-,-) cannot predict (Chapter 3) Notes: 1. H(P) is simply {L: if (HALT(P,P)) then goto L} 2. H uses HALT(-,-) as a subroutine. 3. H(P) halts iff HALT(P,P) returns false iff P(P) does not halt. 4. Let input P be H  H(H) halts iff H(H) does not halt.  HALT is not a correct implementation!

  24. 1.5 Set operations • union, intersection, difference, complement • Definitions 1. A È B = {x | x ∈ A or x ∈ B} 2. A Ç B = {x | x ∈ A and x ∈ B} 3. If A Ç B = {} => call A and B disjoint 4. A - B = {x | x ∈ A but x ∉ B} 5. ~A = U - A • Venn diagram representations • Ex: U = {1,...,10}, A = {1,2,3,5,8} B = {2,4,6,8,10} => A È B , A Ç B , A - B , ~A =?

  25. Set identities (P. 124, 6th edition) (Table 1)

  26. Prove set equalities 1. Show that ~(A È B) = ~A Ç ~B by show that • 1. ~(A È B) Í ~A Ç ~B • 2. ~A Ç ~B Í ~(A È B) Pf: 1. (By definition and logic reasoning)Let x be any element in ~(A È B) . Then x  ~(A È B) iff ~(x  A È B) iff x  A and x  B iff x  ~A and x  ~B. 2. Show (1) by using set builder and logical equivalence. ~(A È B) = {x | x  A È B} = {x | x  ~A and x  ~B} = {x | x ~A} Ç {x | x  ~A} = ~A Ç ~B.

  27. Membership Table 3. Show distributive law: AU(B⋂C) = (AUB)⋂(AUC) by using a membership table. Let x be any element. Then we need only consider 8 cases as to whether x is a member of A or B or C. In all cases, we find that x AU(B⋂C) iff x(AUB)⋂(AUC). Hence the equality holds.

  28. Set equality reasoning 4. Show ~(A È (B Ç C)) = (~C È ~B) Ç ~A by set identities. Pf: ~(A È (B Ç C)) = ~A Ç ~(B Ç C) = ~(B Ç C) Ç ~A = (~B È ~C) Ç ~A = (~C È ~B) Ç ~A

  29. Generalized set operations Def. 6 • A1,A2,...,An: n sets • B = {A1,A2,...,An} 1. A1È A2È An = ÈB = È{i=1,..n} Ai =def ? 2. A1Ç A2Ç An = ÇB = Ç{i=1,..n} Ai =def ? quiz: if B = {} => ÈB =?; ÇB=? • Venn diagram representation of A1È A2È A3 and A1Ç A2Ç A3 . • Principles of inclusion and exclusion Example: A1 = {0,2,4,6,8}, A2 = {0,1,2,3,4}, A3 ={0,3,6,9} => • A1È A2È A3=? • A1Ç A2Ç A3 = ?

  30. Set representation (in computer) • Unordered list or array • union, intersection, ~: time consuming. • Bit string: • U = {a1,...,an} is the universal set. • A: any subset of U • A can be represented by the string: s(A) = x1 x2 x3 .... xn where xi = 1 if a1 ∈ A and 0 otherwise. • fast set operations (e.g. union and intersection) • suitable only if U is not large • constant size representation

  31. 1.6 Functions • Def. 1 [functions] A, B: two sets 1. A partial function f from A to B is a set of pairs (x,y) ∈ AxB s.t., for each x ∈ A there is at most one y ∈ B s.t. (x,y) ∈ f. 2. f is a (total) function if for each x ∈ A there is exactly one y ∈ B with (x,y) ∈ f. 3. If (x,y) ∈ f, we write f(x) = y. 4. f: A  B means f is a function from A to B. • Def. 2 If f: A  B then 1. A: the domain of f; 2. B: the codomain of f If f(a)=b then 3. b is the image of a 4. a is the preimage of b 5. range(f) = ? 6. preimage(f) = ?

  32. Types of functions • Def 4. f A x B ; S: a subset of A, T: a subset of B 1. f(S) =def ? {y  B | x  S with (x,y)  f} 2. f-1(T) =def ? {x  A | y  T with (x,y)  f} • Def. [1-1, onto, injection, surjection, bijection] f: A -> B. • f is 1-1 (an injection) iff? • f is onto (surjective, a surjection) iff? • f is 1-1 & onto <=> f is bijective (a bijection, 1-1 correspondence)

  33. Properties of functions • If there is an onto mapping from A to B, then there is a 1-1 mapping from B to A. • If there is a 1-1 mapping from A to B then there is an onto mapping from B to A. • The onto (≥) and the 1-1 (≤) relations between sets are all preorders (i.e., reflexive and transitive) and are converse to each other. (skip) • If there is a 1-1 mapping from A to B and a 1-1 mapping from B to A, then there is a 1-1 and onto mapping from A to B. pf: (1) Let f: AB an onto. For each b ∈ B, let g(b) = {x ∈ A | f(x) = b} ≠ ∅ ⇒ the function h: BA defined by h(b) = any x ∈ g(b) is 1-1. (2,3) Similar to (1). (4) is hard butunderstandable (reference).

  34. FIGURE 1 (2.3) FIGURE 1 Assignment of Grades in a Discrete Mathematics Class.

  35. FIGURE 2 (2.3) FIGURE 2 The Function f Maps A to B. P. 134

  36. FIGURE 3 (2.3) FIGURE 3 A One-to-One Function. P. 137

  37. FIGURE 4 (2.3) FIGURE 4 An Onto Function. P. 138

  38. FIGURE 5 (2.3) FIGURE 5 Examples of Different Types of Correspondences. P. 139

  39. Real valued functions • F: A  B is a real valued function iff A and B are subsets of R. • f1, f2: real valued functions => 1. f1+f2 (x) =? 2. f1• f2 (x) = f1(x) x f2(x). 3. f is increasing iff? 4. f is strictly increasing iff?

  40. Operations on functions • A, B, C: any sets; f: A  B; g: B  C, then 1. [identity function] idA: A  A s.t. idA(x) = x for all x in A. 2. [composition of g and f] gf: A  C s.t. gf(x) = g(f(x)) for x in A. 3. [inverse] If f is a bijection, then f-1: B  A s.t. f-1(y) = x iff f(x) = y for y in B. • Graphical representations

  41. FIGURE 6 (2.3) FIGURE 6 The Function f －1 Is the Inverse of Function f. P. 139

  42. FIGURE 7 (2.3) FIGURE 7 The Composition of the Functions f and g. P. 141

  43. Properties of operations on functions • f: A  B; g: B C; h: C D, then 1. idA f = f = f idB 2. f(gh) = (fg) h -- associative 3. (fg)-1 = g-1 f-1 If f: A  A is a bijection then 4. f f-1 = f-1f = idA. • Sequences: finite A-sequence -- a: [n] (or [1,n])  A w A-sequence -- a: N (or N+)  A (infinite) B indexed A-sequence -- f: B  A

  44. The growth of functions (Section 3.2, 6th edition) • Summation rules • S ai =def a1 + a2 + ... + an+ (...) • a + (a+d) + (a+2d) +... =? sol: let S = a + (a + d) + (a + 2d) + ... + (a + (n-1)d). Then S = (a + (n-1)d) + (a + (n-2)d) + … + (a + d) + a 2S = (2a + (n-1)d) + …. + (2a + (n-1)d) = (2a + (n-1)d) x n. => S = [a + (n-1)d] n/2 = (a1 + an) n/2. • a + ar + arr + ... = sol: let S = a + ar + ar2 +... + a r (n-1). Then rS = ar + ar2 + ar3 +… arn (1-r)S = a - arn. => S = a(1-rn)/(1-r). (This is valid only if r is not equal to 1.)

  45. FIGURE 10 (2.3) FIGURE 10 Graphs of the (a) Floor and (b) Ceiling Functions. P. 143

  46. TABLE 1 (2.3) + + + + + P. 144

  47. TABLE 1 (2.4) P. 153

  48. TABLE 2 (2.4) P. 157

  49. The growth of functions • If f is a positive valued function, then what do the following terms mean? • O(f) means? • W(f) means? • Q(f) means? • These are useful in analyses of the efficiency of computer algorithms.

  50. Problem, algorithm and complexity • A problem is a general question: • description of parameters [input] • description of solutions [output] • description of how the outputs are related to the inputs. • An algorithm is a step by step procedure to get the related output (i.e., answer) from the given input. • a recipe • a computer program • We want the most efficient algorithm • fastest (mostly) • most economical with memory (sometimes) • expressed as a function of problem input size

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