Section Sets and Set Operations

Transcription

1 Section Sets and Set Operations Definition: A set is a well-defined collection of objects usually denoted by uppercase letters. Definition: The elements, or members, of a set are denoted by lowercase letters. Set Notations: 1. Roster Notation: Lists each element between braces Example 1: 2. Set-builder Notation: A rule is given that describes the property an object x must satisfy to qualify for membership in the set. Example 2: Notation: If a is an element of a set A, we write a A. If a doesn t belong to A we write a / A. Example 3: Let A = {1,2,3}. Definition: Two sets A and B are equal, written A = B, if and only if they have exactly the same elements. (Note: The elements do NOT have to be in the same order.) Example 4: Let A = {1,2,3}, B = {2,1,3}, C = {1,2,3,4} Definition: If every element of a set A is also an element of a set B, then we say that A is a subset of B and write A B. Definition: A is a proper subset of B if A is a subset of B but A does not equal B. (i.e. We write A B if A B and there exists at least one element in B that is not in A) Example 5: Let A = {1,2,3}, B = {2,1,3}, C = {1,2,3,4} Note:, represent containment between sets. To show an element is part of a set, we use. Example 6: If A = {1,2,3,4} determine whether each of the following is true or false: a) 1 A b) 1 A c) {1} A d) {1} A e) {1} A 1

2 Definition: The set that contains no elements is called the empty set and is denoted by /0 OR {} (not {/0}). The empty set is a subset of every set. Example 7: List all subsets of A = {a,b,c} Definition: A universal set, U, is the set of all elements of interest in a particular matter. We have different universal sets for different problems. We use Venn Diagrams to visually represent sets. The Universal Set U is denoted by a rectangle. Subsets of U are represented by circles inside the rectangle. Set Operations: 1. If U is a universal set and A is a subset of U, then the set of all elements in U that are NOT in A is called the complement of A, denoted A c. Example 8: A c = {x U x / A} 2. The intersection of sets A and B is the set of all elements that belong to both A and B. Example 9: A B = {x U x A and x B} 3. The union of sets A and B is the set of all elements that belong to A or B. Example 10: A B = {x U x A or x B } 2

3 Set Complementation: U c = /0 /0 c = U (A c ) c = A A A c = U A A c = /0 Example 11: Let s check some of the above properties using Venn Diagrams. Definition: Two sets A and B are disjoint if A B = /0. Example 12: Let A = {1,3,5,7}, B = {2,4,6,8}. Are A and B disjoint? Properties of Set Operations: Commutative Law: Associative Law: Distributive Law: A B = B A A B = B A A (B C) = (A B) C A (B C) = (A B) C A (B C) = (A B) (A C) A (B C) = (A B) (A C) 3

4 DeMorgan s Laws (A B) c = A c B c (A B) c = A c B c Example 13: Let s prove DeMorgan s Law using Venn Diagrams Example 14: Let A, B, and C be subsets of a universal set U. Shade the following regions on a Venn Diagram: a) A B C c b) A B C c c) (A B) c C d) (B C) A c 4

5 Example 15: Let U = {1,2,3,4,5}, A = {1,2,3}, B = {1,3,5}. Find a) A c b) A B c) A A c d) A A c e) (A B) c f) A B c Example 16: Let U denote the set of all cars in a dealer s lot and A = {x U x is equipped with automatic transmission} B = {x U x is equipped with air conditioning} C = {x U x is equipped with side air bags} Find an expression in terms of A, B, and C for each of the following sets: a) The set of cars with at least one of the given options. b) The set of cars with automatic transmission and side air bags but no air conditioning. c) The set of cars with exactly one of the given options. Section 6.1 Highly Suggested Homework Problems: 9, 11, 15, 19, 21, 29, 31, 35, 37, 41, 47, 49, 61, 63, 67, 69, 71 5

6 Section The Number of Elements in a Finite Set Definition: For a set A, we denote the number of elements in A as n(a). Example 1: If A = {a,b,c}, B = {x x is a letter in the English Alphabet}, and C = /0, find: a) n(a) b) n(b) c) n(c) You will need to remember the following formula: Example 2: Does the formula make sense? n(a B) = n(a) + n(b) n(a B) Example 3: In a recent survey of 200 members of a local sports club, 100 members indicated that they plan to attend the next Summer Olympic Games, 60 indicated that they plan to attend the next Winter Olympic Games, and 40 indicated that they plan to attend both games. How many members of the club plan to attend a) At least one of the two games? b) The Summer Olympic Games only? c) Exactly one of the games? d) None of the games? 6

7 Example 4: If n(a) = 12,n(B) = 12,n(A B) = 5,n(A C) = 5,n(B C) = 4,n(A B C) = 2, n(a B C) = 25, and n(a c B c C c ) = 7, find n(c). Example 5: To help plan the number of meals to be prepared in a college cafeteria, a survey was conducted, and the following data were obtained: 8 students ate only breakfast. 80 students ate only lunch. 96 students ate exactly 2 meals. 68 students ate breakfast and lunch. 58 students ate all three meals. 100 students did not eat dinner. 112 students ate breakfast and dinner. 99 students ate exactly 1 meal. How many students were surveyed? 7

8 Example 6: A survey was conducted of College Station residents to determine what activities they participated in during the 4th of July weekend. It was found that 955 residents watched fireworks. 50 residents only went swimming. 528 residents participated in exactly two of these activities residents watched fireworks or ate BBQ. 250 residents only watched fireworks and ate BBQ. 60 residents did not watch fireworks and did not eat BBQ. 425 residents watched fireworks, ate BBQ, and went swimming. 523 residents went swimming and ate BBQ. How many residents did not eat BBQ? Section 6.2 Highly Suggested Homework Problems: 3, 5, 7, 11, 15, 17, 21, 25, 29, 31, 35, 37 8

9 Section Multiplication Principle Example 1: A yogurt shop has 3 flavors (Chocolate, Vanilla, and Strawberry) and two sizes (Large and Medium). How many different yogurts can be ordered? Multiplication Principle: Suppose a task T 1 can be performed in N 1 ways, a task T 2 can be performed in N 2 ways,..., and a task T n can be performed in N n ways. Then, the number of ways of performing the tasks T 1, T 2,..., T n in succession is given by the product: Example 2: A coin is tossed 3 times, and the sequence of heads and tails is recorded. Determine the number of outcomes. Example 3: An auto manufacturer has 3 different subcompact cars in the line. Customers selecting one of these cars have a choice of 3 engine sizes, 4 body styles, and 3 color schemes. How many different selections can a customer make? 9

10 Example 4: How many three-letter words that have exactly one vowel can be made using the first seven letters of the alphabet where using a letter twice is permitted but having two consonants next to each other is not? Example 5: How many five-digit numbers can be formed if a) Zero is not the first digit? b) Zero cannot be the first digit and no digit can be repeated? c) Zero cannot be the first digit, no digit can be repeated, and each number formed must be even? 10

11 Example 6: Jack and Jill and 5 of their friends go to the movies. They all sit next to each other in the same row. How many ways can this be done if a) there are no restrictions? b) Jill must sit in the middle? c) Jill sits on one end of the row and Jack sits on the other end of the row? d) Jack, Jill, or John sit in the middle seat? e) Jack, Jill, and John sit in the middle seats? f) Jack and Jill must sit next to each other? g) Jill must not sit next to Jack? Section 6.3 Highly Suggested Homework Problems: 1, 3, 5, 7, 9, 11, 15, 19, 21, 23, 25, 27, & Counting Handout 1 11

12 Section Arrangements and Combinations Example 1: Suppose we want to seat 12 people in a row of 12 seats. How many arrangements are possible? The above product is called a factorial: n! = n(n 1)(n 2) Note: 0! = 1 Example 2: How many ways can we select 5 people from a group of 12 and arrange them in 5 chairs? Definition: If we have n distinct elements and we want to take r of them in an arrangement, we say that the number of arrangements of n things taken r at a time is: Example 3: How many ways can we select 25 people from a group of 35 and arrange them in 25 chairs? Arrangement of n objects, not all distinct: Given a set of n objects in which n 1 are alike of one kind, n 2 are alike of another,..., n r alike of another so that n 1 + n n r = n then the number of arrangements of the n objects taken n at a time is: Example 4: Suppose we have 2 identical red marbles, 3 identical green marbles, and 1 blue marble. If we want to line the marbles up in a row, how many distinguishable arrangements of the 6 marbles are there? Example 5: How many distinguishable arrangements can we make from the letters in the word Mississippi? 12

13 Definition: The number of combinations of n items taken r at a time is: Example 6: Example 7: Suppose a high school choir made of 11 students decides to send 2 members to a duet competition. a) How many pairs are possible? b) If it is decided that one particular member is to go, how many different pairs are possible? c) If there are 3 boys and 8 girls in the choir, how many pairs will include at least one boy? 13

14 Example 8: Suppose we have a bag containing 6 different purple candies, 3 different red candies, and 7 different green candies. You choose 5 pieces at random. a) How many samples of 5 candies can be chosen? b) How many samples are there in which all the candies are green? c) How many samples are there in which they are all red? d) How many samples are there in which there are 2 purple and 1 red? e) How many samples are there in which there are no purple candies? f) How many samples contain at least 1 purple? g) How many samples contain exactly 2 purple or exactly 2 green candies? 14

15 Example 9: Suppose we are playing the lottery in which we must choose 6 from 50 numbers. a) How many different lottery picks could we choose if the order we choose our numbers in does not matter? b) How many ways are there to choose no winning numbers? c) How many ways are there to choose at least 3 winning numbers? Example 10: In how many ways can a committee be formed with a chair, a secretary, a treasurer, and four additional people if they are to all be chosen from a group of ten people? Section 6.4 Highly Suggested Homework Problems: 3, 13, 19, 33, 35, 37, 39, 45, 47, 49, 51, 53, 55, 63, 67, 68, 69, 71, 73, & Counting Handouts 2 &3 15