Chapter 2 Sets and Counting

Transcription

1 Chapter 2 Sets and Counting Sets Learn the following: Definition of a set, subset, element of a set, null set, cardinal number of a set, set notations, operations on sets (union, intersection, difference, complement), Venn Diagram, survey problems, De-Morgan s rules, multiplication principle, construction process of license plate, telephone number, different outfits, Permutations (arrangements) and combinations (committee, group, election, selection). Definition Set A set is a collection of well-defined and well-distinguished elements of any nature. If a, b, c, d, e are the elements of a set A then we write A { a, b, c, d, e } Subsets For two sets A and B if all elements of the set B are also the elements of A then B is called the subset of A and we write B A or A B. Null Set A set containing no element in it is called a null set or void set or empty set. Two sets are called mutually exclusive or disjoint if they do not have common elements. Cardinal Number The total number of elements in set is called its cardinal number. For the set A { a, b, c, d, e }, the cardinal number is 5 and we write na ( ) 5 Operations on sets For two sets A and B we have the following operations: Union the union for two sets A and B denoted by A Bis the set of all elements in A or in B. Suppose that x A B then either x A,or x Bor x belongs to both A and B. Intersection the intersection for two sets A and B denoted by A Bis the set of all elements in A and in B. Suppose that x A B then x A and x B. For mutually exclusive sets C and D we have C D {} Difference For two sets A and B, x A\ B is a set of all elements in A but not in B. Complement of a set Suppose U is a universal set and A is its subset. The set U \ A A is the set with all elements in U but not in A is the complement of A. In this case A is the proper subset of U. Notice that AU \ the null set. Cardinal number in a set which is a union of two sets: n( A B) n( A) n( B) n( A B ) Firoz 1

2 Venn Diagram The pictorial representation of elements of sets is called a Venn Diagram (after the name of an English logician John Venn ). We will discuss more on Venn diagram in the class. Examples: 1. Given U { a, b, c, d, e, f }, A { b, c}, B { c, d, e }. Find i) ( A B ) ii) A B 2. List all subsets of E { a, b, c }; find also the cardinal number of the set of all such subsets. 3. Given n( A) 10, n( A B) 29, n( A B ) 5, find nb ( ) 4. Survey questions: use Venn diagram a) A department store surveyed 428 shoppers, and the following information was obtained: 214 made a purchase, and 299 were satisfied with the service they received. If 52 of those who made a purchase were not satisfied with the service, how many shoppers did the following? i) made a purchase and were satisfied with the service 162 ii) made a purchase or were satisfied with the service 351 iii) were satisfied with the service but did not make a purchase 137 iv) were not satisfied and did not make a purchase 77 b) The given data indicates that on average, out of every 100 people in the United States, 40 have the A molecule, 15 have the B molecule, and 45 have neither the A nor the B molecule (O type). What percent of the U.S. population have the following blood type i) Type O ii) Type A iii) Type B iv) Type AB, where AB means A and B c) A consumer survey was conducted to examine patterns in ownership of personal computers, cellular telephones, and DVD players. The following data were obtained: 213 people had personal computers, 294 had cellular telephones, 337 had DVD players, 109 had all three, 64 had none, 198 had cell phones and DVD players, 382 had cell phone or computers, and 61 had computers and DVD players but no cell phones. i) What percent of the people surveyed owned a computer but no DVD player or cell phone? ii) What percent of the people surveyed owned a DVD but no computer or cell phone? Introduction to Combinatorics The fundamental concept of counting, permutations and combinations collectively are called Combinatorics. Firoz 2

3 The fundamental counting principle If you have two T-shirts, two pairs of shoes and three pairs of jeans, how many different outfits can you have? To answer this question one may use the following tree diagram Start T-shirt 1 T-shirt 2 Shoes 1 Shoes 2 Shoes 1 Shoes 2 One can easily find that there are 8 possible outfits available namely T-shirt 1, Shoes 1, (which is one outfit). Another choice would be like T-shirt 1, Shoes 1, etc. Using multiplication principle we can see that the total number of outfits = Thus the principle is summarized below: The total number of possible outcomes of a series of decisions is found by multiplying the number of choices for each decision as follows 1. Draw a box for each decision 2. Enter the number of choices for each decision in the appropriate box and multiply Examples: 1. Five different mathematics books are to be arranged on a desk. How many different arrangements are possible? Answer 120 Solution: We make five boxes as follows and insert possible choices in each case. The books cannot be repeated. There are 5 choices for the first box, 4 choices for the second box and so on. The total number of choices equals Firoz 3

4 2. How many license plates consisting of 2 letters followed by 2 digits are possible? Answer How many different license plates can be made using 2 letters followed by 4 digits, if 2 4 a. Letters and digits may be repeated. Answer b. Letters may be repeated, but digits are not repeated. Answer c. Neither letters nor digits are repeated. Answer Students log on to the California Virtual In Campus with a user name consisting of eight characters: four upper case letters followed by four digits. 4 4 a. How many user names are theoretically possible? Answer b. How many users names for this system have no matching adjacent letters 3 3 or digits? Answer Find the number of 7-digit telephone numbers a. With no repeated digits (lead 0 is allowed). Answer b. With no repeated digits (lead 0 is not allowed). Answer c. With repeated digits allowed including a lead 0. Answer Each student at State University has a student I. D. number consisting of four digits (the first digit is nonzero and digits may be repeated) followed by three of the letters A, B, C, D, and E (letters may not be repeated). How many different student numbers are possible? 3 Answer Factorials: In example 1, we have seen that the total number ways that 5 books can be put in a shelf is which can also be written as 5! General form n! n( n 1)( n 2) Note that 0! 1. Proof is kept as an exercise. Hint: Use n! n( n 1)! where n is a positive integer. 7. Simplify 19852!. Your calculator will fail, but you don t ! 198! 8. Simplify 196!3! n! 9. Find when 10 ( n 1)! Firoz 4

5 Permutations: When more than one item is selected (without replacement) from a single category, and the order of selection is important, the various possible outcomes are called permutations. If you are using TI calculator, you go to MATH PRB 2 for n P r. To find a permutation of 3 different object out of 5 is P(5,3). Using calculator you type 5 MATH PRB 2 3. Your calculator will return a result 60. Try. General formula for permutations: If r items are selected form n different items (no repetition, order is important) n! P( n, r) n Pr ( n r)! Combinations: When more than one item is selected (without replacement) from a single category, and the order of selection is not important, the various possible outcomes are called combinations. If you are using TI calculator, you go to MATH PRB 3 for n C r. To find a combination of 3 different object out of 5 is C(5,3). Using calculator you type 5 MATH PRB 2 3. Your calculator will return a result 10. Try. 1. How many ways in a group of three people have different birthdays? Assume there are 365 days in a year. Answer P (365,3) How many ways are there to seat 5 people in 8 chairs? Answer P (8,5) From 5 faculty members, a committee of 2 is to be formed. In how many ways can this be done? Answer From 6 men and 5 women, a 5-member committee is to be formed. In how many ways this can be formed if no more than 3 women are in the committee? Hint: C(6,5)+C(6,4)C(5,1)+C(6,3)C(5,2)+C(6,2)C(5,3) Firoz 5

6 Quiz 1 Feb 17, 2010 First name Last name 1. Given U { a, b, c, d, e, f }, A { b, c}, B { c, d, e }. Find i) ( A B ) ={ ii) A B={ 2. Construct appropriate example for the sets A, B and U to show that n( A B) n( A B ) 3. A consumer survey was conducted to examine patterns in ownership of personal computers, cellular telephones, and DVD players. The following data were obtained: 213 people had personal computers, 294 had cellular telephones, 337 had DVD players, 109 had all three, 64 had none, 198 had cell phones and DVD players, 382 had cell phone or computers, and 61 had computers and DVD players but no cell phones. Find the following: 1) What percent of the people surveyed owned a computer but no DVD player or cell phone? Firoz 6

7 2) What percent of the people surveyed owned a DVD but no computer or cell phone? Homework 2 on Sets Due on Monday Feb 22, 2010 Your name:.. 1. Let A {1,3,5,8} B {3,5,7}, C {2, 4,6,8}, find the following: a) ( A B ) b) A B c) A\ B d) A\ B A 2. If the universal set is U {1,2,3,4,5,6,7,8,9} and A {1,3,5,7,9} then find U \ A U and ( U\ A ) 3. Use appropriate examples to prove that n( A B) n( A) n( B) n( A B ) 4. Prove logically the DeMorgan s laws: ( A B) A B and ( A B) A B 5. In a consumer survey of 500 people, 200 indicated that they would be buying a major appliance within next month, 150 indicated that they would buy a car, and 25 said that they would purchase both a major appliance and a car. How many will purchase neither? How many will purchase only a car? How many will purchase either a car or a major appliance? What percent of the consumers exactly purchased only one (either a car or a major appliance)? Firoz 7