Topic 1: The Fundamental Counting Principle

Set TheoryTopic 3: Intersection and UnionI can organize information such as collected data and number properties, using graphic organizers, and explain the reasoning.I can explain what a specified region in a Venn diagram represents, using connecting words (and, or, not) or set notation.I can determine the elements in the intersection or the union of two sets.I can identify and correct errors in a solution to a problem that involves sets.If a card is drawn from a standard deck of cards, it will be from one of four suits: clubs (C), spades (S), hearts (H), or diamonds (D).

Try this example in your workbook before looking at the answers on the next 2 slides.Explore

1. a) How many cards are hearts (n(H))?b) How many cards are clubs (n(C))?c)How many cards are hearts or clubs (n( ))?d)Explain how you could use your answers for (a) and (b) to answer (c). Explain why this works.

e)Draw a Venn diagram showing cards that are hearts or clubs.

131326You can simply add them together. This is because there is no overlap!HCSExplore

2. a) How many cards are diamonds (n(D))?b) How many cards are face cards (n(face))?c)How many cards are hearts or clubs (n( ))?d)Explain how you could use your answers for (a) and (b) to answer (c). Explain why this works.

e)Draw a Venn diagram showing cards that are diamonds or face cards.

131222You need to add the number of diamonds and the number of face cards, and subtract the number of cards that are a part of both groups (a diamond face card) so that these are not counted twice!DFDFExplore

InformationThe intersection of sets is the set of elements that are common to two or more sets.In set notation, means the intersection of sets A and B.

The union of sets is the set of all elements in two or more sets.In set notation, means the union of sets A and B.

InformationThe number of elements in the union of two sets can be calculated using the Principle of Inclusion and Exclusion. This principle solves for the union of A and B by adding the number of elements in A to the number of elements of B, and then subtracting the number of elements that are common to both (intersection), since we would have counted those twice.

In set notation, A\B means in set A but not in set B.

Most of the time, you can do these questions using the Venn Diagram. You will not have to use this formula.Example 1In an Alberta school, there are 14 grade 12 students. Of these students, 6 play volleyball and 7 play basketball. There are 3 students who do not play either sport. The following Venn diagram represents the set of students.Explaining the different regions of a Venn diagramB (basketball)S (all grade 12 students)Try this example in your workbook before looking at the answers on the next slide.a) Are the set of volleyball players and the set of basketball players disjoint? Explain how you know.

b) Describe how you can determine the number of students who play volleyball only, basketball only, and both volleyball and basketball? Example 1: Solutions

When you add up the elements, there are more than 14. Some elements, then, must be counted twice.Determine how many too many we have when we add them all up. This is the number of students that are in the overlapping section.

6 + 7 + 3 = 16Since there are only 14 students, we know that 2 students were counted twice.2c) How many students play only volleyball (n(V\B)) ? How many people play only basketball (n(B\V))?

d) Place all numbers in the Venn diagram.Example 1: Solutions

Since 2 students play both sports, 4 play volleyball only (6-2) and 5 play basketball only (7-2).2543e) Identify what each part of the Venn diagram represents. i. Describe the union of B and V. Determine n(BUV).

ii. Describe the complement of the union of B and V. Determine n(BUV).

iii.Describe the intersection of B and V. Determine n(BV).

Example 1: Solutions

2543All students that play volleyball, baseball, or both. n(BUV) = 11.All students that do not play volleyball, baseball, or both. n(BUV) = 3.All students that play volleyball and baseball. n(BV) = 2.f) Is n(B) + n(V) = n(BUV) a correct statement. Explain.Example 1: Solutions

25437 + 6 = 13NO! This is not a true statement, since you will have counted the students who play both sports twice. Once in n(B) and once in n(V).Example 2Jamaals first survey was on January 4th, where he surveyed 34 people at his gym. He learned that 16 people do weight training three times a week, 21 people do cardio training three times a week, and 6 people do neither.Using reasoning to determine the number of elements in a setExample 2a) Megan added 16, 21, and 6 to get 43. Is this correct? Explain why or why not?Using reasoning to determine the number of elements in a setJamal only surveyed 34 people, so Megans calculation must be incorrect.

Megan didnt account for the fact that 9 of the 16 and 9 of the 21 are contained in the overlapping section. In essence, Megan counted these items twice.Example 2b) Draw the Venn diagram that represents this data. Write the number of people in each region in the diagram.

9127GymCardioWeights16 9 = 7There are 16 people that do weight training, 9 of which are in the overlapping section.21 9 = 12There are 21 people that do cardio training, 9 of which are in the overlapping section.6There are 6 people that do neither.Example 2Jamaals second survey was on April 4th, where he surveyed 27 people at his gym. He learned that 10 people do weight training three times a week, 14 people do cardio training three times a week, and 3 people do neither.c) Draw the Venn diagram that represents this data.Using reasoning to determine the number of elements in a set1410GymCardioWeightsSince 10+14+3 = 27 (the total number surveyed) there is no overlap!3There are 3 people that do neither.Example 2Using reasoning to determine the number of elements in a setandorExample 3Morgan surveyed the 30 students in her math class about their eating habits. She found that 18 ate breakfast, and 5 of those 18 also ate lunch. There were 3 students who ate neither breakfast nor lunch.

How many students ate lunch?

Correcting ErrorsExample 3Tyler solved this problem, as shown below, but made an error. What error did Tyler make? [Hint: Try solving yourself first.]

Correcting Errors

Example 3: Solution

Tyler forgot to take 5 away from 18 when putting the number in the left circle. The whole circle must equal 18!Need to KnowEach element in a universal set appears only once in a Venn diagram.

You can count the number of elements in a universal set by counting the elements in each region.

Need to KnowEach area of a Venn diagram represents something different.

Need to KnowThe union of two or more sets, AUB, has the following characteristics:is indicated by the word or.it consists of all elements that are in at least one set.its represented by the entire region of the sets in a Venn diagram.The intersection of two or more sets, AB, has the following characteristics:is indicated by the word and.it consists of all elements that are common to all sets.it is represented by the region of overlap of the sets in a Venn diagram.

Need to KnowThe number of elements in the union of sets A and B can be calculated in two ways:

using the Principle of Inclusion and Exclusion:

(In a Venn diagram, this prevents the elements in the region of overlap from)

using

Need to KnowIf sets A and B are disjoint sets, the following is true: n(AUB) = n(A) + n(B) because n(AB) = 0

Youre ready! Try the homework from this section.