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Section 2.4

Math in Our World. Section 2.4. Using Sets to Solve Problems. Learning Objective. Solve problems by using Venn diagrams. Using Venn Diagrams with Two Sets. Step 1 Find the number of elements that are common to both sets and write that number in region II.

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Section 2.4

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  1. Math in Our World Section 2.4 Using Sets to Solve Problems

  2. Learning Objective • Solve problems by using Venn diagrams.

  3. Using Venn Diagramswith Two Sets 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 but not in B and write that number in region I. Repeat for the elements in B but not in A, 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. U A B I II III IV

  4. 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.

  5. 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). U L C I II III 2 34 8 IV 6

  6. 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.

  7. 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 41 + 79 + 31 = 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. U P T I II III 79 41 31 IV 349

  8. Classwork p. 80-81: 1, 2, 3, 4, 15

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