1. UNIT 3

2. OUTLINE: Sets and Subsets Set Operations the Laws of Set Theory Counting and Venn Diagrams. A First Word on Probability. The Axioms of Probability. Self Learning Exercise: Conditional Probability: Independence.

3. 3.1 Sets and Subsets A well-defined collection of objects (elements or members) e.g., set of even numbers Terminologies: finite sets, infinite sets, cardinality of a set, subset Representing a set A={1,3,5,7,9} B={x|x is odd} C={1,3,5,7,9,...} cardinality of A=5 (|A|=5)

5. 3.1 Sets and Subsets set equality subsets

6. A is a proper subset of B. C is a subset of B. Subset and proper subset

7. Example

9. Theorem 1

11. Example

12. 3.1 Sets and Subsets null set or empty set : {},  universal set, universe: U power set of A: the set of all subsets of A A={1,2}, P(A)={ , {1}, {2}, {1,2}} If |A|=n, then |P(A)|=2 n.

14. Theorem 2

15. 3.1 Sets and Subsets common notations (a) Z=the set of integers={0,1,-1,2,-1,3,-3,...} (b) N=the set of nonnegative integers or natural numbers (c) Z + =the set of positive integers (d) Q=the set of rational numbers={a/b| a,b is integer, b not zero} (e) Q + =the set of positive rational numbers (f) Q*=the set of nonzero rational numbers (g) R=the set of real numbers (h) R + =the set of positive real numbers (i) R*=the set of nonzero real numbers (j) C=the set of complex numbers

16. 3.1 Sets and Subsets common notations (k) C*=the set of nonzero complex numbers (l) For any n in Z +, Z n ={0,1,2,3,...,n-1} (m) For real numbers a,b with a<b, closed interval open interval half-open interval

17. 3.2 Set Operations and the Laws of Set Theory Def. 3.5 For A,B a) b) c) union intersection symmetric difference Def.3.6 mutually disjoint Def 3.7 complement Def 3.8 relative complement of A in B

20. Theorem 3

22. For any universe U and any set A,B in U, the following statements are equivalent: a) b) c) d) reasoning process Theorem 4

24. 3.2 Set Operations and the Laws of Set Theory The Laws of Set Theory

25. 3.2 Set Operations and the Laws of Set Theory The Laws of Set Theory

27. 3.2 Set Operations and the Laws of Set Theory s dual of s (s d ) Theorem 3.5 (The Principle of Duality) Let s denote a theorem dealing with the equality of two set expressions. Then s d is also a theorem.

28. Example

29. 3.2 Set Operations and the Laws of Set Theory Ex. 3.17 What is the dual of Since

30. Problem

31. Negate

32. U A A A B Venn diagram

33. Proof using Venn diagram

34. Proof using Membership Table

35. more problems

36. 3.3 Counting and Venn Diagrams Ex. 3.23. In a class of 50 college freshmen, 30 are studying BASIC, 25 studying PASCAL, and 10 are studying both. How many freshmen are studying either computer language?

37. Example Given 100 samples set A: with D 1 set B: with D 2 set C: with D 3 Ex 3.24. Defect types of an AND gate: D 1 : first input stuck at 0 D 2 : second input stuck at 0 D 3 : output stuck at 1 with |A|=23, |B|=26, |C|=30,, how many samples have at least one defect? How many have no defect?

38. Solution Ans:57 & 43

39. example There are 3 games Tennis, FootBall, Cricket. In how many ways can one play one game each day so that one can play each of the three at least once during 5 days? set A: without playing Tennis set B: without playing FootBall set C: without playing Cricket

40. Definition

41. Exercise problems: 3.3 Ans: 55

42. Ans: 40

43. Ans: a)12b)2c)16

44. Problem : A survey of 500 television viewers of a sports channel produced the following information: 285 watch cricket, 195 watch hockey, 115 watch football, 45 watch cricket and football, 70 watch cricket and hockey, 50 watch hockey and football and 50 do not watch any of the three kind. (i) How many viewers in the survey watch all three kinds of games? (ii) How many viewers watch exactly one of the sports. Ans: 20, 325

45. Problem: Professor Diane gave her chemistry class a test consisting of three questions. There are 21 students in her class and every student answered at least one question. Five students did not answer the first question, seven failed to answer the second question and six did not answer the third question. If nine students answered all three questions, how many answered exactly one question? Answer 6

46. 3.4 A Word on Probability U=sample space (a set of all outcomes for each situation) event A Pr(A)=the probability that A occurs=|A|/|U|, where A is a set of events a elementary event Pr(a)=|{a}|/|U|=1/|U|

47. Example

50. Ex. 3.27 If one tosses a coin four times, what is the probability of getting two heads and two tails? Ans: sample space size=2 4 =16 event: H,H,T,T in any order, 4!/(2!2!)=6 Consequently, Pr(A)=6/16=3/8 Each toss is independent of the outcome of any previous toss. Such an occurrence is called a Bernoulli trial.

51. Problem In a class of 20 students, two students are selected in random for CR. In how many ways can the selection be made. Suppose Anil and Bhavani are two students of this class, then what is the probability that Both Anil and Bhavani are selected Niether Anil nor Bhavani are selected Only Anil and not Bhavani is selected

52. Solution

53. Problem

54. solution

55. Problem

56. PROBLEMS

57. Solution

58. Problem WYSIWYG (What You See Is What You Get) How many different arrangements of this acronym are there? How many have consecutive W’s and Y’s? What is the probability that the arrangements have both consecutive W’s and Y’s? Find the probability that the a random arrangement of these seven letters starts and ends with W.

59. Solution The probability that the arrangement starts and ends with W

60. The axioms of probability

61. Problem If two dices are rolled simultaneously, what is the probability that the following events occur A: Rama rolls 6 ( top faces of the dice sum to 6) B: the sum of the dice is atleast 7 C: Rama rolls an even sum D: the sum of the dice is <=6

62. Solution

63. But

65. Problem

66. solution

68. Problem

69. Solution

71. Problem

72. Solution

74. Problem

75. Solution