1. Part I The Essence of Sets 1 ENM 503 Sets rd

2. 1000 Lockers rd 2 There are 1000 lockers and all are opened. Then I go by each and reverse the state. If open, I close it; if closed, I open it. Then I repeat for every 2 lockers, then every 3 lockers, etc. When done, what are the states of the lockers? What are you modeling mathematically?

3. Chord of Circle rd 3 Express the length L of a chord of circle with radius r as a function of x being the distance from the center. L = 2(r 2 - x 2 ) 1/2 Find length of chord in circle of radius 13 that is 5 units from the center. L = 2(169 – 25) 1/2 = 24. r L/2 x

4. Sum of Odd Numbers are Squares 4 1 = 1 2 1 + 3 = 2 2 1 + 3 + 5 = 3 2 1 + 3 + 5 + 7 = 4 2 1 + 3 + 5 + 7 + 9 = 5 2 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 = ? 1+3+5+7+9+11+13+15+17+19+21+23+25+27+29+31 +33+35+37+39+41+43+45+47+49+51+53+55+57+59 +61 = ? rd

5. Odds and Cubes 5 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 … rd

6. Square Arrays of the Odds 6 1 3 5 add => 3 2 7 9 11 13 15 17trace = 3 3 sum all 3 4 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 rd

7. Sum of cubes of the numbers in set equal square of sum 7 Choose any number; 10; list its divisors DivDivisors#its cube 111 1 21,22 8 5 1,52 8 10 1,2,5,10 4 64 9 81 You try. Pick any number and repeat above. rd

8. Fibonacci Sequence Recursive – up a staircase one or two steps at a time with n steps n # of ways 1 1 22: 11, 2 33: 111,12, 21 45: 1111, 112, 121, 211, 22 f n = f n-1 + f n-2 For n = 5 steps, take 1 step 4  5 ways 11111,1112,1121,1211,122 thereafter or 2 steps 3  3 ways 2111, 212, 23 thereafter yielding 5 + 3 = 8 rd 8

9. Some Properties of Fibonacci Sequence rd 9 (fibo-lst 1 22)  (1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 10946) Which terms are evenly divisible by 3, 5, 8, 13 and 55? Observe: 3 2 = 5*2 -1; 5 2 = 8*3+1; 8 2 = 13*5 -1; … 1 2 +1 2 +2 2 = 2*3; 1 2 +1 2 +2 2 +3 2 = 3*5; 1 2 +1 2 +2 2 +3 2 +5 2 = 5*8; … 3 2 +5 2 = 34; 8 2 + 13 2 = 233; 13 2 +21 2 = 610; …

10. Numerology rd 10 How much wheat can be put on a chessboard with 1 grain on the first square, 2 on the next, 4 on the third etc.? S = 1 + 2 + 4 + 8 + 16 + 32 + … + 2 63 = (1 – 2 64 ) / (1 – 2) = 18,446,744,073,709,551,615 grains of wheat Roughly a train reaching a thousand times around the Earth filled with wheat.

11. Pythagorean Triples rd 11 mnm 2 – n 2 2mnm 2 + n 2 32 512 13 61 3512 37 65 1160 61 7 6 1384 85 8 7 15112 113 6011 34791320 3721 = 61 2 8413 68872184 7225 = 85 2 10 6 64 = 4 3 120 136 6 3 27 = 3 3 36 45 Try one yourself by picking an m > than n.

12. Asking for a Raise rd 12 Would you rather receive a raise in salary of $300 every 6 months or $1000 every year?

13. Set Theory 13 Theory: A formal mathematical system consisting of a set of axioms and the rules of logic for deriving theorems from those axioms. Set theory – a branch of abstract mathematics set – a concept so basic that it is an undefined term consider a set a well-defined collection of objects that are called the elements of the set rd

14. Limits of Sequences rd 14 1. Write first 5 terms of sequence {1 + 1/n} and express the limit of the sequence. 2 3/2 4/3 5/4 6/5 … 1 2. Express limit of sequence 0.9 0.99 0.999 0.9999 … 1 – 1/10 n …

15. Means rd 15 Arithmetic m = (a + b)/2 Geometric m = (ab) ½ Harmonic m = 2ab/(a + b) Weighted Mean of a and n with weights x and y (ax + by)/(x + y)

16. The origins of sets 16 Georg Ferdinand Ludwig Philipp Cantor, b. Mar. 3, 1845, d. Jan. 6, 1918, was a Russian-born German mathematician best known as the creator of SET THEORY (Peano also) and for his discovery of the transfinite numbers. He also advanced the study of trigonometric series, was the first to prove the non-denumerability of the real numbers, and made significant contributions to dimension theory (Cardinality) Pushed for abstraction rd

17. Cantor Mathematics 17 Is mathematics Discovered or Invented or Neither? Cantor was an inventor; his invention was set theory. Different sizes of infinity (Cardinality) {1 2 3 …} aleph null symbol  0 countably infinite  +  =  Real numbers are unaccountably infinite  1 rd

18. 18 Euclid alone has looked on Beauty bare Sonnet from The Harp-Weaver and Other Poems by Edna St. Vincent Millay (1923) Euclid alone has looked on Beauty bare. Let all who prate of Beauty hold their peace, And lay them prone upon the earth and cease To ponder on themselves, the while they stare At nothing, intricately drawn nowhere In shapes of shifting lineage; let geese Gabble and hiss, but heroes seek release From dusty bondage into luminous air. O blinding hour, O holy, terrible day, When first the shaft into his vision shone Of light anatomized! Euclid alone Has looked on Beauty bare. Fortunate they Who, though once only and then but far away, Have heard her massive sandal set on stone. rd

19. Goldbach's Conjecture 19 Goldbach asked Euler "Is every even number > 2 a sum of two primes? Remains unsolved today. rd

20. Fermat's Last Theorem x n + y n = z n 20 Find integers x, y and z greater than 1 and integer n > 2 for which x n + y n = z n. "I have found a remarkable proof. The margin is too small to contain it." Fermat None exist. Proved by Andrew Wiles in 1994 rd

21. What is a set? 21 A set is a collection of things. And a collection? A set of things. Each entry in a set is known as an element. Sets are written using brackets { } with their elements listed in between For example the English alphabet could be written as: {a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z} even numbers could be {0,2,4,6,8,10,...} (Note: the dots at the end indicating that the set goes on infinitely, but beware of …) rd

22. Set Equality 22 Definition (Equality of sets): Two sets are equal if and only if they have the same elements. More formally, for any sets A and B, A = B if and only if  x  x  A, then x  B. Thus for example {1, 2, 3} = {3, 2, 1}, that is the order of elements does not matter, and {1, 2, 3} = {3, 2, 1, 1}, that is duplications do not make any difference for sets. A = {1, 2, 3}; B = {1, {2 3} }; C = {{1, 2} 3} 1 is an element of A, and B, but not C (note:  reads “for all” and  reads “such that”) rd

23. Subsets 23 Definition (Subset): A set A is a subset of a set B if and only if everything in A is also in B. More formally, for any sets A and B, A is a subset of B, and denoted by A  B, if and only if  x  A, then x  B If A  B, and A  B, then A is said to be a proper subset of B and it is denoted by A  B. For example {1, 2}  {3, 2, 1}. Also {1, 2}  {3, 2, 1}. rd

24. A Little Theorem 24 If A  B and B  C, then A  C A little proof: 1.  x  A, then x  B definition of subset 2.  x  B, then x  C definition of subset 3. Therefore  x  A, then x  C rd

25. More to do with subsets 25 If even one element of one set is not contained within the other, then they are not subsets. If A were defined as {1,2,3,4,5} and B as {3,4,5,6} then B would not be a subset of A since 6  B but 6  A. The symbol for “not a subset” is . We would write B  A. C = {1 2 {3 6 9}) contains 3 elements 3  C` rd

26. Set of Numbers 26  The empty set { } N = {0 1 2 3 … } N* = {1 2 3 … } exclude 0 Z = {… -3 -2 -1 0 1 2 3 … } Q = {set of a/b with a, b integers} the rational numbers R set of real numbers C set of complex numbers Normal sets are sets that do not contain themselves is the set of all birds in the set? rd

27. Some Examples Number of Elements 27 b  {a, b, c, d} 4 e  {a, b, c, d} 4 {1, 2, 4, 5} = {2, 1, 5, 4} 4 {1, 2, 3}  {1, 2, 3} 3 {1, 2, 3}  {1, 2, 3, 4} 3 < 4 {1, 2, 3}  {1, 2, 3, 4} 3 < 4 {1, 2, 3,..., 1000} is a finite set. 1000 {1, 2, 3,...} is an infinite set.  {1 1 2 3 5 3 3} = {1 2 3 5}4 E =  0 F = {  }1 rd

28. Two sets of note: 28 The set containing zero elements is the “empty” or “null” set and is denoted by { } or  or nil. All sets under consideration are regarded as subsets of a fixed set known as the Universal set and denoted by U rd

29. Set of All set Russell's Paradox 29 Set of all dogs is not a dog, but Consider the set of all sets. It is a member of itself? Normal sets - sets that do no contain themselves as elements Non-normal sets – sets that contain themselves as s elements The set of all cats is Normal; the set of all things that are not cats is Non-normal The set of all sets is also a set –Non-normal Let S = {x | x  x}. Is S  S? If S  S, then S  S; if S  S, then S  S. rd

30. Berry's Paradox"The smallest positive integer not definable in under eleven words.positiveinteger 30 "Since there are finitely many words, there are finitely many phrases of under eleven words, and hence finitely many positive integers that are defined by phrases of under eleven words. Since there are infinitely many positive integers, this means that there are positive integers that cannot be defined by phrases of under eleven words. By the well ordering principle, if there are positive integers that satisfy a given property, then there is a smallest positive integer that satisfies that property; therefore, there is a smallest positive integer satisfying the property "not definable in under eleven words". This is the integer to which the above expression refers. The above expression is only ten words long, so this integer is defined by an expression that is under eleven words long; it is definable in under eleven words, and is not the smallest positive integer not definable in under eleven words, and is not defined by this expression. This is a paradox: there must be an integer defined by this expression, but since the expression is self-contradictory (any integer it defines is definable in under eleven words), there cannot be any integer defined by it.finitelywell ordering principle rd

31.  is a subset of every set 31 The empty set is a subset of every set and every set is a subset of itself. Denumerable – can be put into a 1-1 correspondence with the natural numbers (aleph null  0 ) rd

32. Venn Diagrams 32 Venn Diagrams were first developed by John Venn in the 1880s. They are useful for illustrating the relationships between elements in a set. For example if we want to represent the set of all counting numbers, and illustrate how even numbers and multiples of 3 are related, we could draw the following picture: 6 12 18 24 … The Universe rd

33. John Venn 33 John Venn (August 4, 1834 – April 4, 1923), was a British mathematician, who is famous for conceiving the Venn diagrams, which are used in many fields, including set theory, probability, logic, statistics, and computer science. A Diagram of Venn rd

34. More of Venn’s Diagrams 34 Two sets having no elements in common A subset and equal sets rd

35. Another hero - George Boole 35 In 1854 he published An investigation into the Laws of Thought, on Which are founded the Mathematical Theories of Logic and Probabilities. Boole approached logic in a new way reducing it to a simple algebra, incorporating logic into mathematics. He pointed out the analogy between algebraic symbols and those that represent logical forms. It began the algebra of logic called Boolean algebra which now finds application in computer construction, switching circuits etc.Boolean algebra Born: 2 Nov 1815 in Lincoln, England Died: 8 Dec 1864 in Ballintemple, Ireland rd

36. Boolean Algebra 36 A Boolean algebra is an algebra in which the binary operations are chosen to model mathematical or logical operations in Set Theory. For any sets A and B, a Boolean algebra is defined under the operations of union, intersection and complement. In abstract algebra, a Boolean algebra is an algebraic structure (a collection of elements and operations on them obeying defining axioms) that captures essential properties of both set operations and logic operations. Specifically, it deals with the set operations of intersection, union, complement; and the logic operations of AND, OR, NOT, XOR, NAND, & NOR gates rd

37. Boole's Contribution 37 Replace logical statements by statements about sets n is a prime number implies a set of prime numbers with n in the set. If P and Q are logical statements, then P implies Q is equivalent to P  Q In Boolean algebra a + bc = (a + b)(a + c) = aa + ac + ba + bc U Q P = a + ac + ba + bc = a(1 + c) + b(a + c) = a + ba + bc = (1 + b)a + bc = a + bc rd

38. Union 38 A union of two or more sets is another set that contains everything contained in the previous sets. Union is designated by the symbol  or  If A and B are sets then A  B represents the union of A and B The union of A and B is the set of all elements that are either in A or B (or both), therefore A + B = {x | x  A or x  B}. “OR” logic rd

39. Claude Shannon & Boolean Algebra 39 A B C D 0 0 0 0 0 0 1 0 0 1 0 1 0 1 1 1 1 0 0 0 1 0 1 0 1 1 0 1 A 1 1 1 1 B D C D = aBc + aBC + ABc + ABC = aB(c + C) + AB(c + C) = aB + AB = (a + A)B = B rd

40. You try one. 40 Input Output A B C D 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 0 1 0 0 0 1 0 1 1 1 1 0 0 1 1 1 1 rd

41. XOR, NOR, NAND Gates 41 XOR NOR NAND a b Outa b Outa b Out 0 0 00 0 10 0 1 0 1 10 1 00 1 1 1 0 11 0 01 0 1 1 1 0 1 1 01 1 0 rd

42. Inclusive Or vs. Exclusive Or Inclusive or Exclusive OR? You may go to college at Harvard or Yale. Knowledge of Basic or Fortran is needed for the job Price includes dessert or drink To enter the country, you need a passport or a driver’s license 42 rd

43. Translating English Sentences p = It is above freezing q = It is raining” It is above freezing and it is snowing It is above freezing but not snowing It is not above freezing and it is not snowing It is either snowing or above freezing (or both) If it is above freezing, it is also snowing It is either above freezing or it is snowing, but it is not raining if it is above freezing It is above freezing is necessary and sufficient for it to be raining. pqpq P~qP~q ~p~q P+q P → q (p+q)(p → ~q) P ↔ q 43 rd

44. Examples of the union of two sets 44 Example 1: A = {1,2,3,4,5}; B = {5,7,9,11,13} A  B = {1,2,3,4,5,7,9,11,13} Example 2: A = {all the books written by Charles Dickens} B = {all the books written by Mark Twain} A  B = {all books written by either Charles Dickens or Mark Twain} rd

45. Boolean Algebra 45 a + b = b or a commute a * 0 = 0; a * 1 = a identity a * (b + c) = a*b + a*cdistributive a + a' = 1complement a*a' = 0 0' = 1; 1' = 0 (a')' = a (1 + b) * (a + 0) = aDuality (0 * b) + (a * 1) = a x * x = x 2 = xIdempotent x + x = x rd

46. Simulation On each face-down card in a deck of 52 is a randomly selected positive integer. If the last card you turn over when you stop is the largest in the deck, you win. You can stop at any time. You win $50 but it costs $10 to play. Are you game? (setf ss (swr 52 (upto 10000))) Split deck in two and turn over first half and find the maximum. Then stop on the second half as soon as one is found exceeding the max of the first half. Soul Mate? rd 46

47. Intersection 47 The intersection of two (or more) sets is those elements that they have in common. Intersection is designated by the symbol  or by juxtaposition. So if A and B are sets then the intersection is denoted by A  B or AB. The intersection of A and B is the set of all elements that are common to A and B, therefore AB = {x | x  A and x  B} “And” logic rd

48. Examples of the intersection of two sets 48 Example 1 A = {1,3,5,7,9}; B = {2,3,4,5,6} The elements in common are 3 and 5 AB = {3,5} Example 2 A = {The English alphabet} B={vowels} So AB = {vowels} Example 3 A = {1,2,3,4,5}; B = {6,7,8,9,10} In this case A and B have nothing in common. AB =  is called the “empty or null set.”  = { } rd

49. Mutual Exclusive Sets 49 Two sets are mutually exclusive (also called disjoint) if they do not have any elements in common; they need not together comprise the universal set. The following Venn diagram represents mutually exclusive (disjoint) sets. AB =  rd

50. Complement 50 Given the set A, then the set A c is the complement of A consisting of all elements not in A; i.e. A c = {x| x  A} A AcAc Let U = universal set (the set of all objects under discussion) Then A + A c = U and A A c =  (they are mutually exclusive) N(A) = N(S) – N(A c ) to indicate the number in a set. The complement of a set A is written as A c U rd

51. Inclusion-Exclusion 51 N(A + B) = N(A) + N(B) – N(AB) A c B c = (A+B) c by DeMorgan's law N(A c B c ) = N(A+B) c = N – N(A + B) N(A c B c ) = N – N(A + B) = N – N(A) – N(B) + N(AB) N is cardinality of the Universe set. The complement of the union of two sets is the intersection of the complements. The complement of the intersection of two sets is the union of the complements rd

52. Set Cardinality The cardinality of a set is the number of elements in a set, written as |A| Examples Let A = {1, 2, 3, 4, 5}. Then |A| = 5 |  | = 0 Let S = { , {a}, {b}, {a, b} c}. Then |S| = 5 52 rd

53. Disjoint sets Sets are disjoint if their intersection is the empty set {1, 2, 7} and {3, 4, 5, 6} disjoint {a, d} and {3, 4} are disjoint  and  are disjoint even though    Because the intersection is . 53

54. Genie Commands 54 (setf x '(1 3 5 7 8) y '(2 3 6 7 8)) (set-difference x y)  (1 5) (set-union x y)  (1 2 3 5 6 7 8) (set-intersection x y)  (3 7 8) (set-exclusive-or x y)  (1 5 2 6) (member 7 y)  (7 8) rd

55. A Partition 55 A partition of a set S is a subdivision of S into subsets which are disjoint and whose union is S. That is, each x  S belongs to one and only one of the subsets If A 1, A 2, …, A n form a partition, then A 1 + A 2 + … + A n = S, and for any A i and A j i  j, A i A j =  Often the set S = U Partitions of integers: Find the partitions of the number 5. 5; 1 4; 2 3; 1 1 3; 1 2 2; 1 1 1 2; 1 1 1 1 1 rd

56. Partitions 56 A partition is a group of identical objects dividing the group into unordered subsets of various sizes. List the 7 partitions of 5 1+1+1+1+1, 1 + 2 + 2, 2 + 3, 5, 1 + 1 + 1 + 2, 1 + 1 + 3, and 1 + 4. rd

57. The four disjoint regions of two intersecting sets forming a partition 57 The four regions into which a Venn diagram with two circles divides the universal set can be identified as intersections of the two subsets and their complements as labeled in the following Venn diagram. U = (A c B c ) + (AB c ) + (AB) + (A c B) rd

58. Three intersecting sets 58 rd

59. SET THEOREMS 59 If A  B and B  C then A  C If A  B then A  B = A. If A  B then A  B = B. AB  A  (A + B) rd

60. Examples 60 Let S be the set of all integers, and let A = {2, 4, 6, 8} B = {5, 6, 7, 8} C = {positive even integers} D = {1, 2, 3}. Then A + B = {2, 4, 5, 6, 7, 8} AB = {6, 8} AC = A C c = {0, 1, -1, -2, 3, -3, -4, 5, -5,...} A(B + C) = A rd

61. Closure 61 If A and B are any two sets then A + B is a set AB is a set A c is a set rd

62. The Laws of the Algebra of Sets 62 De Morgan's Laws: (A  B)' = A'  B‘ (A  B)' = A'  B‘ Idempotent Laws: (A  A) = A (A  A) = A Associative Laws: (A  B)  C = A  (B  C) (A  B)  C = A  (B  C) Commutative Laws: (A  B)= (B  A) (A  B) = (B  A) Distributive Laws: A  (B  C) = (A  B)  C) A  (B  C) = (A  B)  C) Identity Laws (A   ) = A (A  U) = U (A   ) =  (A  U) = A Complement Laws (A  A’) = U (A’)’ = A (A  A’) =  U’ = ,  ’ = U rd

63. Augustus De Morgan 63 (June 27, 1806 - March 18, 1871) was an Indian-born British mathematician and logician. He formulated De Morgan's laws and was the first to introduce the term, and make rigorous the idea of mathematical induction. De Morgan crater on the Moon is named after him. His main accomplishments were in the field of logic, although he studied both probability and algebra. He wrote a series of textbooks on arithmetic, algebra, trigonometry, calculus, complex numbers, probability, and logic. He is known for the development of De Morgan’s Laws, which deal with the logic of relations. rd

64. Set Algebra – an example 64 Simplify: rd

65. Cartesian Product 65 The Cartesian product of two sets, A and B, is the set of all ordered pairs (a, b) with a  A and b  B. A x B = { (a, b) | a  A, and b  B }. A x B is the set of all ordered pairs whose first component is in A and whose second component is in B. rd

66. Cartesian Product - examples 66 1. If A = {a, b} and B = {1, 2, 3}, then A x B = { (a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3) }. 2. Let S = {H, T}. (H stands for Heads, T stands for Tails) S x S = { (H,H), (H,T), (T,H), T,T) }. If S is the set of outcomes of tossing a coin once, then S x S is the set of outcomes of tossing a coin twice. rd

67. Yet another example 67 S = {1, 2, 3, 4, 5, 6} The set of outcomes of rolling a die S x S = {(1, 1),(1, 2),(1, 3),(1, 4),(1, 5),(1, 6), (2, 1),(2, 2),(2, 3),(2,4), (2, 5),(2, 6),(3, 1),(3, 2),(3, 3),(3, 4),(3, 5),(3, 6),(4, 1),(4, 2), (4, 3),(4, 4),(4, 5),(4, 6),(5, 1),(5, 2),(5, 3),(5, 4),(5, 5),(5, 6), (6, 1),(6, 2),(6, 3),(6, 4),(6, 5),(6, 6)} rd

68. A Cartesian Product Theorem 68 If A, B, and C are any sets, then (A + B)C = AC + BC Quick student exercise: Demonstrate the truth of this theorem by creating an example. That is, show both sets have the same ordered pairs. Another quick student exercise: Demonstrate that A(BC) = (AB)(AC) is true or not true. rd

69. Power Sets 69 Let U be the universal set. Then the set whose elements are all the subsets in U is called the power set of U and is denoted by P U. There are 2 n elements in the set P U where n is the number of elements in U (power-set (upto 5))  ((1 2 3 4 5) (1 3 4 5) (1 2 4 5) (2 3 4 5) (1 2 3 4) (1 2 3 5) (2 3 4) (1 4 5) (2 4 5) (3 4 5) (2 3 5) (1 2 3) (1 2 5) (1 3 4) (1 3 5) (1 2 4) (2 5) (2 4) (3 5) (4 5) (3 4) (1 4) (1 5) (1 3) (1 2) (2 3) (5) (4) (2) (1) (3) NIL) (length *)  32 rd

70. Power Set (Reals) > |Power Set| 70 Power Set of a set > cardinality of set Let S = {1 2 3 4 5} of 5 elements Then the Power set of S is (power-set (upto 5))  ((1 2 3 4 5) (1 3 4 5) (1 2 4 5) (2 3 4 5) (1 2 3 4) (1 2 3 5) (2 3 4) (1 4 5) (2 4 5) (3 4 5) (2 3 5) (1 2 3) (1 2 5) (1 3 4) (1 3 5) (1 2 4) (2 5) (2 4) (3 5) (4 5) (3 4) (1 4) (1 5) (1 3) (1 2) (2 3) (5) (4) (2) (1) (3) NIL) (length *)  32 Thus the power set of the Reals cannot be put into a 1- 1 correspondence with the Reals. rd

71. The Mandatory Example 71 Let U = {a,b,c,d} (powerset '(a b c d)) Then P U = { , {a}, {b}, {c}, {d}, {a,b}, {a,c}, {a,d}, {b,c}, {b,d}, {c,d}, (a,b,c} {a,b,d}, {a,c,d}, {b,c,d}, {a,b,c,d} } rd

72. More of the mandatory example 72 Assume that {a,b,c,d} represent a committee of 4 members, each having a single vote where a simple majority vote is required. The each element of P U can be viewed as a voting coalition. Then W = { (a,b,c}, {a,b,d}, {a,c,d}, {b,c,d}, {a,b,c,d} } is the set of winning coalitions and W’ = { , {a}, {b}, {c}, {d}, {a,b}, {a,c}, {a,d}, {b,c}, {b,d}, {c,d}} is the set of non-winning coalitions. L = { , {a}, {b}, {c}, {d} }  W’ is a losing coalition (its complement is a winning coalition), and B = {{a,b}, {a,c}, {a,d}, {b,c}, {b,d}, {c,d} }  W’ is a blocking coalition. rd

73. A Quick Review 73 A set is a collection of items, referred to as the elements of the set. x  A means that x is an element of the set A. x  A means that x is not an element of the set A. B = A means that A and B have the same elements. B  A means that B is a subset of A; every element of B is also an element of A. B  A means that B is a proper subset of A; in other words, B  A, but B  A. Ø is the empty set, the set containing no elements. It is a subset of every set. A finite set is a set that has finitely many elements. An infinite set is a set that does not have finitely many elements. rd

74. Part II - Why do we care? The Application of Sets – a look ahead ENM 503 Sets Part II 74 rd

75. Some Uses of Sets Circuit Theory Logic Decision-making bodies (voting) Binary arithmetic Computer design Probability theory Risk analysis Reliability Fault-tolerant Software A set of Allen wrenches A tea set 75 rd

76. Let’s solve a problem Twenty-four dogs are in a kennel. Twelve of the dogs are black, six of the dogs have short tails, and fifteen of the dogs have long hair. There is only one dog that is black with a short tail and long hair. Two of the dogs are black with short tails and do not have long hair. Two of the dogs have short tails and long hair but are not black. If all of the dogs in the kennel have at least one of the mentioned characteristics, how many dogs are black with long hair but do not have short tails? B = 12; ST = 6; BSTLH = 1 BST~LH = 2; ~BSTLH = 2 BLH~ST = ? 76 rd

77. This solution is for the dogs 9 - x + 2 + 1 + 1 + 2 + x + 12 - x = 24 27 - x = 24 x = 3 77 rd

78. Modeling Feasible Regions (optimization) Consider the following sets where x = production level for product A and y = production level for product B A = {(x,y)| x >= 0 }; B = {(x,y)| y >= 0 }; C = {(x,y)| y <= 4 }; D = {(x,y)| x + y <= 5 }; E = {(x,y)| x <= 3 }. Then ABCDE = x y 4 3 x + y = 5 78 rd

79. Modeling Feasible Regions – 2 A = {(x,y)| x >= 0 }; B = {(x,y)| y >= 0 }; C = {(x,y)| y <= 4 }; D = {(x,y)| x + y <= 5 }; E = {(x,y)| x <= 3 }; F = {x,y| x,y are integers} Then ABCDEF = {(0,0), (0,1), (0,2), (0,3), (0,4), (1,0), (1,1), (1,2), (1,3), (1,4), (2,0), (2,1), (2,2), (2,3), (3,0), (3,1), (3,2)} x y 4 3 x + y = 5 79 rd

80. Disjoint Feasible Regions A = {(x,y)| x >= 0 }; B = {(x,y)| y >= 0 }; C = {(x,y)| x + y = 5 } Feasible region = A  B  (C  D) x y 4 x + y = 4 x + y = 5 80 rd

81. Modeling Random Processes Example 1 Let S = a sample space, the outcome from tossing a pair of coins S = {(H,H), (H,T), (T,H), (T,T)} Let E 1 = a random event (outcome), obtaining two heads E 1 = {(H,H)} Let E 2 = a random event (outcome), obtaining at least one head E 2 = {(H,H),(H,T),(T,H)} 81 rd

82. Modeling Random Processes Example 2 Let S = the set of all possible outcomes from selecting three parts from a parts bins containing both defective and non-defective parts. Isomorphism: (toss a fair coin 3 times} S = {(N,N,N), (N,N,D), …, (D,D,D} What is the size of the sample space? 2 * 2 * 2 = 8 Let E = the event, at least one part is defective E = {(N,N,D), (N,D,N), (D,N,N), (N,D,D), (D,N,D), (D,D,N), (D,D,D)} Let F = the event, exactly two parts are defective F = {(N,D,D), (D,N,D),(D,D,N)} 82 rd

83. Modeling Random Processes Example 3 Random process: select at random a freshman from the UD fall 2011 class to deliver a keynote address Then the sample space S = the set of all freshmen Let A = the random event, a male is selected Let B = the random event, a student having a GPA of 3.5 or better selected Let C = the random event, a student from Ohio is selected Then the event D, a male having a GPA below 3.5 and not from Ohio is given by: D = AB c C c 83 rd

84. M Yet Another Example Consider a random process in which a manufactured product is selected at random. Products coming off the assembly line may or may not meet design specifications. In addition, these products may or may not pass final inspection. Let S = set of all products coming off assembly Let M = the event, product meets specification Let P = the event, product passes inspection. P S 84 rd

85. Continuing the Example Then S = MP + M’P + MP’ +M’P’ ( a partition) where MP) = the event, the product meets specs and passes inspection, M’P = the event, the product does not meet specs and passes inspection, MP’ = the event, the product meets specs and does not pass inspection, M’P’ = the event, the product does not meet specs and does not pass inspection M P S 85 rd

86. Risk Analysis: Fault Trees Analysis (FTA) Graphical design technique Perspective on faults or failures Model events as sets Focus on a catastrophic event (top event) Top-down deductive analysis 86 rd

87. Fault Tree Symbols – Logic Gates AND gate - a logic gate where an output event occurs only when all the input events have occurred (i.e. an intersection of events) symbol:  OR gate - a logic gate where an output event occurs if at least one of the input events have occurred (i.e. a union of events) symbol:  87 rd

88. More Fault Tree Symbols - Events Resultant event - a fault event resulting from the logical combination of other fault events and usually an output to a logic gate. Basic event - an elementary event representing a basic fault or component failure. Incomplete event - an event that has not been fully developed because of lack of knowledge or its unimportance. 88 rd

89. General Structure of a Fault Tree Top Event System Failure Resultant Events AND/OR gates Basic Events 89 rd

90. Example of AND / OR Gates Let T = the event, tank ruptures O = the event, overpressure W = the event, wall fatigue failure then T = O + W Tank Ruptures OR Overpressure (a) Wall Fatigue Failure 90 rd

91. Example of AND / OR Gates Let E = the event, excessive temperature R = the event, relief valve failure then O = ER Overpressure Excessive Temperature Relief Valve Fails AND (b) 91 rd

92. Example of AND / OR Gates T = O U W = (ER)  W Excessive Temperature Relief Valve Fails AND Tank Ruptures OR Overpressure (c) Wall Fatigue Failure 92 rd

93. Alarm System Example automatic sensor power source backup power alarm switch meter observer 2 ½ Mile Island 93 rd

94. Alarm System Fault Tree T- Alarm Failure A-Power Failure G-Manual Alarm Failure E-Primary power fails F-Backup fails I-Human Error B-Sensor Failure J-Switch fails K-Meter Fails H-Auto Sensor Fails AND OR C-Alarm Failure D Secondary alarm failure 94 rd

95. Boolean representation of Top Event T = A  B  C + D = ( EF )  ( GH )  C + D = ( EF )  [(I + J + K )H] + C + D 95 rd

96. Applications in Reliability A system consists of n components and the sample space is the set of all possible states of the components where the components are either operating or failed E i = the event, component i does not fail, and R = the event, the system does not fail where the system is composed of n components If the components are in series (i.e. if any one component fails the system fails, then R = E 1 E 2  n 1 2n … 96 rd

97. More Reliability An electronic assembly consists of 5 components arranged in a series – parallel circuit as shown: A C D B E Define the events A, B, C, D, and E as the events components A, B, C, D, and E do not fail respectively. The assembly will fail only if there is no operating path from left to right through the network. Express the event F, the assembly does not fail, in terms of the events A, B, C, D, and E. F = (AB + CD)E 97 rd

98. An alternate approach to More Reliability Let S = {A, B, C, D, E}, then define the power set, P S = { , {A}, {B}, {C}, {D}, {E}, {A,B}, {A,C}, {A,D}, {A,E}, {B,C}, {B,D}, {B,E}, {C,D}, (A,B,C}, {A,B,D}, {A,B,E}, {A,C,D}, {B,C,D}, {B,C,E}, {C,D,E}, {A,B,C,D}, {A,B,C,E}, etc. } Then F = {{A,B,C,D,E}, {A,B,E}, {C,D,E}, {B,C,D,E}, {A,C,D,E}, {A,B,C,E} A C D B E 2 5 = 32 elements i.e. set enumeration 98 rd

99. Asking for a Raise rd 99 Would you rather receive a raise in salary of $300 every 6 months or $1000 every year?

100. Polynomial Expansions 100 1 – x m+1 / (1 – x) = 1 + x + x 2 + … + x n 1 + x + x 2 + x 3 + … + x n 1 – x 1 + x + x 2 x 3 + … + x n - x - x 2 – x 3 + … - x n - x n+1 1 - x n+1 - rd