Section 2.4 Using Sets to Solve Problems Math in Our World

Learning Objective  Illustrate set statements involving two sets with Venn diagrams.  Illustrate set statements involving three sets with Venn diagrams.  Solve problems by using Venn diagrams.

Illustrating a 2 Set Statement with a Venn Diagram Region I represents the elements in set A that are not in set B. Region II represents the elements in both sets A and B. Region III represents the elements in set B that are not in set A. Region IV represents the elements in the universal set that are in neither set A nor set B. A IIIIII U B IV

Step 1 Find the number of elements that are common to both sets and write that number in region II. Step 2 Find the number of elements that are in set A and not set B by subtracting the number in region II from the total number of elements in A. Then write that number in region I. Repeat for the elements in B but not in region II, and write in region III. Step 3 Find the number of elements in U that are not in either A or B, and write it in region IV. Step 4 Use the diagram to answer specific questions about the situation. Using Venn Diagrams with Two Sets A IIIIII B U IV

EXAMPLE 1 Solving a Problem by Using a Venn Diagram In 2008, there were 36 states that had some form of casino gambling in the state, 42 states that sold lottery tickets of some kind, and 34 states that had both casinos and lotteries. Draw a Venn diagram to represent the survey results, and find how many states have only casino gambling, how many states have only lotteries, and how many states have neither.

EXAMPLE 1 Solving a Problem by Using a Venn Diagram SOLUTION Step 1 Draw the diagram with circles for casino gambling and lotteries and label each region with Roman Numerals. Step 2 Thirty-four states have both, so put 34 in the intersection of C and L, which is region II. Step 3 Since 36 states have casino gambling and 34 have both, there must be 2 that have only casino gambling. Put 2 in region I. Since 42 states have lotteries and 34 have both, there are 8 that have only lotteries. Put 8 in region III. Step 4 Now 44 states are accounted for, so there must be 6 left to put in region IV. Now we can answer the questions easily. There are only two states that have casino gambling but no lottery (region I). There are eight states that have lotteries but no casino gambling (region III), and just six states that have neither (region IV). IIIIII L IV C U

EXAMPLE 2 Solving a Problem by Using a Venn Diagram In a survey published in the Journal of the American Academy of Dermatologists, 500 people were polled by random telephone dialing. Of these, 120 reported having a tattoo, 72 reported having a body piercing, and 41 had both. Draw a Venn diagram to represent these results, and find out how many respondents have only tattoos, only body piercings, and neither.

EXAMPLE 2 Solving a Problem by Using a Venn Diagram SOLUTION Step 1 Draw the diagram with circles for people with Tattoos and people with body Piercings. Step 2 Place the number of respondents with both tattoos and body piercings (41) in region II. Step 3 There are 120 people with tattoos and 41 with both, so there are 120 – 41, or 79, people with only tattoos. This goes in region I. By the same logic, there are 72 – 41, or 31, people with only piercings. This goes in region III. Step 4 We now have = 151 of the 500 people accounted for, so 500 – 151 = 349 goes in region IV. There are 79 people with only tattoos, 31 with only piercings, and 349 with neither. IIIIII P IV T U

Venn Diagrams with Three Sets Region I represents the elements in set A but not in set B or set C. Region II represents the elements in set A and set B but not in set C. Region III represents the elements in set B but not in set A or set C. Region IV represents the elements in sets A and C but not in set B. Region V represents the elements in sets A, B, and C. Region VI represents the elements in sets B and C but not in set A. Region VII represents the elements in set C but not in set A or set B. Region VIII represents the elements in the universal set U, but not in set A, B, or C.

EXAMPLE 3 Solve a Problem by Using a Venn Diagram A survey of 300 first-year students at a large Midwestern university was conducted to aid in scheduling for the following year. Responses indicated that 194 were taking a math class, 210 were taking an English class, and 170 were taking a speech course. In addition, 142 were taking both math and English, 111 were taking both English and speech, 91 were taking both math and speech, and 45 were taking all three. Draw a Venn diagram to represent these survey results, and find the number of students taking (a) Only English.(b) Math and speech but not English. (c) Math or English.(d) None of these three subjects.

M E S EXAMPLE 3 Solve a Problem by Using a Venn Diagram SOLUTION 300 Students194 Math (M)210 English (E)170 Speech(S) 142 Math and English111 English and Speech 91 Math and Speech45 All Three Step 1 Draw the diagram for 3 sets. Step 2 We know from the given information region V—the number of students taking all three classes. So we begin by putting 45 in region V. Step 3 There are 142 students taking both math and English, but we must subtract the number in all three classes to find the number in region II: 142 – 45 = 97. In the same way, we get 91 – 45 = 46 in region IV (both math and speech) and 111 – 45 = 66 in region VI (both English and speech)

M E S EXAMPLE 3 Solve a Problem by Using a Venn Diagram SOLUTION CONTINUED 300 Students194 Math (M)210 English (E)170 Speech(S) 142 Math and English111 English and Speech 91 Math and Speech45 All Three Step 4 Now we can find the number of elements in regions I, III, and VII. There are 194 students in math classes, but = 188 are already accounted for in the diagram, so that leaves 6 in region I. Of the 210 students in English classes, = 208 are already accounted for, leaving just 2 in region III. There are 170 students in speech classes, with = 157 already accounted for. This leaves 13 in region VII

M E S EXAMPLE 3 Solve a Problem by Using a Venn Diagram SOLUTION CONTINUED 300 Students194 Math (M)210 English (E)170 Speech(S) 142 Math and English111 English and Speech 91 Math and Speech45 All Three Step 5 Adding up all the numbers in the diagram so far, we get 275. That leaves 25 In region VIII. Step 6 Now that we have the diagram completed, we turn our attention to the Questions

M E S EXAMPLE 3 Solve a Problem by Using a Venn Diagram SOLUTION CONTINUED 300 Students194 Math (M)210 English (E)170 Speech(S) 142 Math and English111 English and Speech 91 Math and Speech45 All Three (a)Students taking only English are represented by region III—there are only 2. (b) Math and speech but not English is region IV, so there are 46 students. (c) Students taking math or English are represented by all but regions VII and VIII. So there are only 38 students not taking either math or English, and 300 – 38 = 262 who are. (d) There are 25 students outside of the regions for all of math, English, and speech

9.2 Homework 2.4 wksht ex. 4, 8, 12, 16