Topic 1: The Fundamental Counting Principle

Set TheoryTopic 3: Applications of Set TheoryI 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.ExploreThe 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.

CDPS7625401813Try this explore activity in your workbook before looking through the next slide.You should finda) 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 = 78537 + 4 + 1 + 3 = 154 + 5 = 97 + 4 + 5 + 62 + 18 + 0 = 96InformationThe 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 1Rachel surveyed Grade 12 students about how they communicated with friends over the previous week.Completing a Venn diagram

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 1The Venn diagram below represents the following sets:

1. Complete the following Venn diagram.UExample 1

Used all three forms of communicationTexted and used a social networking site but did not call on a cell phoneCalled on a cell phone and used a social networking site but did not textCalled on a cell phone and texted but did not use a social networking site12764419 12 = 718 12 = 656 12 = 44Example 1

Only called on a cell phone, only texted, or only used a social networking site127644Only called on a cell phone66 44 12 6 = 44Only texted76 44 12 7 = 1313Only used a social networking site34 6 12 7 = 99Example 12. What does the universal set U represent in the situation?The universal set represents all grade 12 students that were surveyed by Rachel. 10c) 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?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?

12764441394 + 44 + 12 + 6 + 13 + 7 + 9 = 95%100 95 = 5%5444 + 13 + 9 = 26%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.

12764441394 + 44 + 13 = 61%Add up the data in the Venn diagram circles but do not include circle S.5

Example 2Shannons 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, Shannons 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.Correcting errors that involve setsTry drawing a Venn diagram that represents this information before looking at the next slide.

Example 2

TCW20 students walk, cycle, and use public transit.2035 students walk and cycle.35 20 = 151580 students walk and use public transit.80 20 = 6060100 students cycle and use public transit.100 20 = 8080370 students use public transit370 80 20 60 = 210210445 students cycle or use public transit445 210 60 20 80 15 = 6060265 students cycle or use public transit265 60 80 20 15 - 60 = 3030a) Draw a Venn diagram to represent this informationExample 2

TCW201560802106030b) How many students use green transportation for travelling to and from school?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)210 + 60 + 20 + 80 + 60 + 15 + 30 = 475210 + 80 + 20 + 60 = 370210 + 80 + 20 + 60 + 60 + 15 = 44560 + 80 + 20 + 15 + 60 + 30 = 265Example 2

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

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 cant just subtract these!Example 3Hillary 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}.

Searching on the internetExample 3a) 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.

Searching on the internet

C = {all sites that contain the words popular and culture}CExample 3b) 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 diagramSearching on the internet

P = {all sites that contain the exact phrase popular culture}PCExample 3Once 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.Searching on the internet

T = {all the sites that contain the exact phrase television shows}PT

This search produced 105000 results.

CExample 4Landon 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.)Searching databases

Example 4Searching databases

Neither HDTV display capability nor artificial intelligence sound controlAlternately:Not HDTV display capability and not artificial intelligence sound control

HDTV display capability and not artificial intelligence sound controlExample 4Searching databasesb) Given the number of elements shown in iv, how many search results would be produced by H U S?Need to KnowTo represent three intersecting sets with a Venn diagram, use three intersecting circles:

ABC is represented by region h. AB is represented by the union of regions e and h. AC is represented by the union of regions g and h. BC is represented by the union of regions h and i.

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.

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 of two or more sets.

Youre ready! Try the homework from this section.