Unit 10:^ Diagram Techn/qtl»

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Problem Solving UnitVENN DIAGRAMS

OBJECTIVES

When you have completed this unit you will:

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1. Be able to state the three basic relationships illustrated in VennDiagrams.

2. Be able to represent information given in a problem or statementby constructing a Venn Diagram.

3. Be able to demonstrate the ability to solve problems by observinginformation represented on Venn Diagrams.

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Unit 7 presents another method of visually representing information in problems.This unit covers Venn Diagrams. The diagrams use circles to represent state-ments or facts presented in a problem. You may be familiar with Venn Diagramsas they are used in Math to show the relationship between sets and their membersor elements.

When using Venn Diagrams to show how facts are related, there are three basicrelationships. The following examples demonstrate these relationships:

A. ALL dogs (A) are animals (B).

B. ! 3 women (A) are Democrats (B).

The shaded area represents women who are Democrats.

C. No (NONE) people (A) are cars (B).

To solve the problems in this unit carefully work your way through the followingCHECKLIST:

1. Read each problem slowly and thoroughly.

2. Determine what are key statements.

3. Look for relationships between statements that imply ALL, SOME, andNONE.

4. Using circles to represent key statements, diagram the relationshipbetween statements.

5. Re-read the problem to be sure that your diagram is an accuraterepresentation of the facts presented.

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EXAMPLE

Make a Venn diagram with 3 circles showing the following relationships.

Some x are y. No x are z. No y are z.

Step 1: The Key Statements are X, Y and Z.

Step 2: Use circles to represent the SOME relationship between X and Y.

Step 3: Show the NONE relationship between X and Z as well as Y and Z(see above).

Problem 1

Make a Venn diagram with 3 circles; show the following relationships.

Some x are y. Some x are z. No y are z.

Problem 2

Make a Venn diagram showing the relationship between cats, animals and cars,using one circle to represent cats, another to represent animals, and a third to rep-resent cars.

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Problem 3

Make a Venn diagram showing the relationship between cats, dogs and animals.

Problem 4

If some of Totteville's residents have brown eyes and some of Totteville's resi-dents are women, is the statement "Some of Totteville's residents are brown-eyedwomen" true, false, or unsubstantiated?

Problem 5

For this problem assume the two statements are correct and make a diagram torepresent the relationships. Then answer the questions.

All bears are butterflies. All bees are bears.

a. Can you be certain that all bees are butterflies?b. Can you be certain that all butterflies are bees?

Problem 6The fire department wants to send booklets on fire hazards to all teachers andhomeowners in town. How many booklets does it need, using these statistics?Use a Venn diagram in solving this problem.

Hint: Send Booklets to Homeowners and Teachers.Some Homeowners are Teachers.

Homeowners 50,000Teachers 4,000Teachers who own their homes... 3,000

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Problem 7

An insurance company wants to contact all physicians and all licensed drivers intown. Using these statistics, how many people must be contacted? Draw a Venndiagram with your solution.

Licensed drivers 8,000Physicians 750Physicians with drivers' licenses 750

Problem 8

The government wants to contact all druggists, all gun store owners, and allparents in a town. How many people must be contacted, using these statistics?

Druggists 10Gun store owners 5Parents 3,000Druggists who own gun stores 0Druggists who are parentsGun store owners who are parents 3

Problem 9

A survey of 100 students gave the following data:

41 students taking Spanish.29 students taking French.26 students taking Russian.15 students taking Spanish and French.8 students taking French and Russian.

19 students taking Spanish and Russian.5 students taking all three languages.

a) Find the number of students among the 100 that are not taking any of the threelanguages, b) How many take just one of the languages? c)The number of stu-dents studying exactly two languages, d) The number of students who take Frenchif and only if they take Russian.

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Problem 10

Among 70 freshman students a survey showed that:

23 were taking physics.25 were taking biology.22 were taking chemistry.

6 were taking physics and biology.7 were taking biology and chemistry.8 were taking chemistry and physics.2 were taking all three sciences.

How many of the freshman were taking none of the three sciences? How manywere taking just one of the three sciences?

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REVIEW

After reviewing the following solutions to Problems 1 through 10, you should beready for the Mastery Demonstration for this unit.

Problem 1:

Problem 2

Some x are y. Some x are z. No y are z.

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Problem 3

Problem 4

From the statement we can't tell whether none of the brown-eyed residents arewomen (as shown in diagram 1) or whether some of the brown-eyed residents arewomen (as shown in diagram 2).

Therefore, the statement is not definitely false (diagram 1) or definitely true(diagram 2). It is just unsubstantiated.

" Tottevilleresidents

A. X \brown-eyedpeople

Diagram 1 Diagram 2

Problem 5

Here is the diagram showing that all bees are bears and all bears are butterflies.

Answer forquestion a.

The diagram shows that all bees are butterflies. Anything inthe circle for bees is automatically in the circle for butterflies.

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Answer forquestion b.

The diagram shows that there can be butterflies which are notbees. For example, the dot in the diagram below is a butterflythat is not a bee.

This is a butterflythat is not a bee.

Problem 6: The Fire Department needs 51,000 booklets.

Homeowners50.000

Teachers4,000

The total booklets needed is: 47,0003,0001.000

51,000

Problem 7: 8,000 people need to be contacted.Seven hundred and fifty licensed drivers are physicians.

Licensed drivers

8,000

Physicians

750

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Problem 8: 3,005 people need to be contacted.

Druggists10

Parents3,000

The total number that must be contacted is:2,990

732

+ 2.3,005

Gun store owners5

Problem 9a) 41 Students are not taking any of the three Languages.b) 27 Students are taking one language.c) 27 Students are taking two Languages.d) 3 Students are taking French if & only if they take Russian.

French

Spanish

Russian

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Problem 10

a) 25 Students are not taking any of the three sciences.b) 22 Students are taking just one of the sciences.

Biology

Chemistry Physics

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