Topic 3: Applications of Set Theory

I can explain how set theory is used in applications such as internet searches, database queries, data analysis, games, and puzzles. I can identify and correct errors in a solution to a problem that involves sets. I can solve a contextual problem that involves sets, and record the solution, using set notation.

Explore… The Venn diagram displays the results of a survey of 100 families regarding technology in their homes. C represents the number of families with a computer, D represents the number of families with a DVD player, P represents the number of families with a smart- phone. S represents all the families surveyed. C D P S 7 62 5 4 18 1 3 Try this explore activity in your workbook before looking through the next slide.

You should find… a) How many families have a computer at home? b) How many families have all three machines in their homes? c) How many families have none of the machines in their homes? d) How many families do not have a smart-phone? e) How many families have a computer and a DVD player? f) How many families have a computer or a smart-phone? 7 + 4 + 5 + 62 = 78 5 3 7 + 4 + 1 + 3 = 15 4 + 5 = 9 7 + 4 + 5 + 62 + 18 + 0 = 96

Information The Principle of Inclusion and Exclusion can be used to determine the number of elements in the union of three sets: Most of the time we can do this by simply looking at a Venn diagram and counting the elements.

Example 1 Completing a Venn diagram Rachel surveyed Grade 12 students about how they communicated with friends over the previous week. 66% called on a cell phone. 76% texted. 34% used a social networking site. 56% called on a cell phone and texted. 18% called on a cell phone and used a social networking site. 19% texted and used a social networking site. 12% used all three forms of communication.

Example 1 The Venn diagram below represents the following sets: U 1. Complete the following Venn diagram.

Example 1 Used all three forms of communication Texted and used a social networking site but did not call on a cell phone 19 – 12 = 7 Called on a cell phone and used a social networking site but did not text 18 – 12 = 6 Called on a cell phone and texted but did not use a social networking site 44 56 – 12 = 44 12 6 7

Example 1 Only called on a cell phone, only texted, or only used a social networking site 44 4 13 Only called on a cell phone 66 – 44 – 12 – 6 = 4 12 6 7 Only texted 76 – 44 – 12 – 7 = 13 9 Only used a social networking site 34 – 6 – 12 – 7 = 9

Example 1 2. What does the universal set U represent in the situation? The universal set represents all grade 12 students that were surveyed by Rachel.

Example 1 3.a) What percentage of people called on a cell phone and texted but did not use a social network site? b) What percent of students who only called on a cell phone, only texted, or only used a social networking site? 44 4 + 13 + 9 = 26% c) What percent of students used at least one of these three forms of communication. Explain your answer. d) What percentage of people use none of these communication methods? 4 + 44 + 12 + 6 + 13 + 7 + 9 = 95% 44 4 13 12 6 7 9 100 – 95 = 5% 5

Example 1 4. Determine the percent of students who called on a cell phone or texted, but did not use a social networking site. Express your result in set notation. 4 + 44 + 13 = 61% Add up the data in the Venn diagram circles but do not include circle S. 44 4 13 12 6 7 9 5

Example 2 Correcting errors that involve sets Shannon’s high school starts a campaign to encourage students to use “green” transportation for travelling to and from school. At the end of the first semester, Shannon’s class surveys the 750 students in the school to see if the campaign is working. They obtain these results: 370 students use public transit. 100 students cycle and use public transit. 80 students walk and use public transit. 35 students walk and cycle. 20 students walk, cycle, and use public transit. 445 students cycle or use public transit. 265 students walk or cycle. Try drawing a Venn diagram that represents this information before looking at the next slide.

Example 2 a) Draw a Venn diagram to represent this information 20 students walk, cycle, and use public transit. T C 35 students walk and cycle. 35 – 20 = 15 80 210 60 80 students walk and use public transit. 80 – 20 = 60 20 60 15 100 students cycle and use public transit. 100 – 20 = 80 30 W 370 students use public transit 370 – 80 – 20 – 60 = 210 265 students cycle or use public transit 265 – 60 – 80 – 20 – 15 - 60 = 30 445 students cycle or use public transit 445 – 210 – 60 – 20 – 80 – 15 = 60

Example 2 b) How many students use green transportation for travelling to and from school? 210 + 60 + 20 + 80 + 60 + 15 + 30 = 475 c) Verify your solution by calculating the following: Number of students that use public transit, n(T) Number of students that cycle or use public transit, n(CUT) Number of students that walk or cycle, n(WUC) T C 80 210 60 210 + 80 + 20 + 60 = 370 20 60 15 210 + 80 + 20 + 60 + 60 + 15 = 445 30 W 60 + 80 + 20 + 15 + 60 + 30 = 265

Example 2 d) How many students use exactly one method of green transportation? 210 + 60 + 30 = 300

Example 2 c) Ali solved this problem as shown below but made some errors. What errors did Ali make? There are a number of people that are in the overlapping section, so you can’t just subtract these!

Example 3 Searching on the internet Hillary and Liam are working on a project for their World Issues class. They need to use the internet to gather information about popular culture, especially social criticism on television. Liam suggests they search for popular culture using a search engine. Hillary and Liam start by defining the universal set as U = {all sites on the internet}.

Example 3 Searching on the internet a) The first search produced 186 000 000 results: The Venn diagram below illustrates the results of this search. As the students perform additional searches, add to this diagram. In set notation, define the results of the first search, set C. C C = {all sites that contain the words popular and culture}

Example 3 Searching on the internet b) They continue to refine their search to narrow down the number of hits. The next search produced 6 520 000 results: In set notation, define the limiting results of the refined search, set P. Add set P to the Venn diagram C P P = {all sites that contain the exact phrase popular culture}

Example 3 Searching on the internet Once they realized that the addition of the quotation marks made their search more specific, they used them when searching for additional pieces. c) The students then added television shows to their search. In set notation, define the part of the search that was new, set T. Add set T to the Venn diagram. C T P T = {all the sites that contain the exact phrase television shows} This search produced 105000 results.

Example 4 Searching databases Landon wants to purchase a new television. He asks the sales associate at Best Buy to perform some searches for him using the store database. Televisions that have HDTV display capability are represented by H, and televisions that have artificial intelligence sound control are represented by S. The universal set U represents all of the televisions in the database. a) Complete the table below. (In the third column, shade in the appropriate region of the Venn diagram.) 𝑯∩𝑺

Example 4 Searching databases 𝑯′ 𝑯∩𝑺′ 𝑯\𝑺 Neither HDTV display capability nor artificial intelligence sound control Alternately: Not HDTV display capability and not artificial intelligence sound control 𝑯′ HDTV display capability and not artificial intelligence sound control 𝑯∩𝑺′   𝑯\𝑺

Example 4 Searching databases b) Given the number of elements shown in iv, how many search results would be produced by H U S? 𝑛 𝐻∪𝑆 =12+5+1=18

Need to Know To represent three intersecting sets with a Venn diagram, use three intersecting circles: A∩B∩C is represented by region h. A∩B is represented by the union of regions e and h. A∩C is represented by the union of regions g and h. B∩C is represented by the union of regions h and i.

Need to Know The Principle of Inclusion and Exclusion can be used to determine the number of elements in the union of three sets: Set theory is useful for solving many types of problems, including internet searches, database queries, data analyses, games, and puzzles.

You’re ready! Try the homework from this section. Need to Know Concepts related to sets can be used to search for websites on the internet: Put an exact phrase in quotation marks. Connect words or phrases with “and” to search for sites that contain both. The word “and” represents the intersection of two or more sets. Connect words or phrases with “or” to search for sites that contain either one or the other, or both. The word “or” represents the union of two or more sets. You’re ready! Try the homework from this section.