1 Vinris

## Presentation on theme: "Topic 3: Applications of Set Theory"— Presentation transcript:

1 Topic 3: Applications of Set Theory

2 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.

3 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.CDPS762541813Try this explore activity in your workbook before looking through the next slide.

4 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?= 7853= 154 + 5 = 9= 96

5 InformationThe 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.

6 Example 1Completing a Venn diagramRachel 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.

7 Example 1 The Venn diagram below represents the following sets:
U1. Complete the following Venn diagram.

8 Example 1 Used all three forms of communication
Texted and used a social networking site but did not call on a cell phone19 – 12 = 7Called on a cell phone and used a social networking site but did not text18 – 12 = 6Called on a cell phone and texted but did not use a social networking site4456 – 12 = 441267

9 Example 1Only called on a cell phone, only texted, or only used a social networking site44413Only called on a cell phone66 – 44 – 12 – 6 = 41267Only texted76 – 44 – 12 – 7 = 139Only used a social networking site34 – 6 – 12 – 7 = 9

10 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.

11 Example 13.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= 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?= 95%4441312679100 – 95 = 5%5

12 Example 14. 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.= 61%Add up the data in the Venn diagram circles but do not include circle S.44413126795

13 Example 2Correcting errors that involve setsShannon’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.

14 Example 2 a) Draw a Venn diagram to represent this information
20 students walk, cycle, and use public transit.TC35 students walk and cycle.35 – 20 = 15802106080 students walk and use public transit.80 – 20 = 60206015100 students cycle and use public transit.100 – 20 = 8030W370 students use public transit370 – 80 – 20 – 60 = 210265 students cycle or use public transit265 – 60 – 80 – 20 – = 30445 students cycle or use public transit445 – 210 – 60 – 20 – 80 – 15 = 60

15 Example 2b) How many students use green transportation for travelling to and from school?= 475c) 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)TC8021060= 370206015= 44530W= 265

16 Example 2d) How many students use exactly one method of green transportation?= 300

17 Example 2c) 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!

18 Example 3Searching on the internetHillary 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}.

19 Example 3Searching on the interneta) The first search produced 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.CC = {all sites that contain the words popular and culture}

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

21 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.CTPT = {all the sites that contain the exact phrase television shows}This search produced results.

22 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

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

24 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

25 Need to KnowTo 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.

26 Need to KnowThe 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.

27 You’re ready! Try the homework from this section.
Need to KnowConcepts 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 oftwo or more sets.You’re ready! Try the homework from this section.

Панк попытался высвободиться и повернуться. - Эдуардо. Это ты, приятель? - Он почувствовал, как рука незнакомца проскользнула к его бумажнику, чуть ослабив хватку.