Venn Diagram and Problem Solving

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Venn Diagram:

Venn Diagram Sir Gerald DG. Banaag Math 7 July 12, 2012

Essential Questions:

Essential Questions How do we know when an object is intrinsically part of a set or not? Why do we have to consider objects as such? Why do you think the study of the concept of sets important? How is the concept of set being used in other fields of study? Which is more essential: the learning of the concept of sets or the acquisition of knowledge of the set of real numbers?

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Use Venn diagrams to represent sets, subsets, and set operations Understand the notation, terminology, and concepts related to Venn diagrams and set notation Solve problems involving sets using Venn diagrams. What You'll Learn

Venn Diagram:

Venn Diagram To draw a Venn diagram, you first draw a rectangle which is called your "universe". In the context of Venn diagrams, the universe is not "everything", but "everything you're dealing with right now". Let's deal with the following list of things: moles, swans, rabid skunks, geese, worms, horses, Edmontosorum (a variety of duck-billed dinosaurs), platypuses, and a very fat cat. We will call our universe "Animals"

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Venn Diagram moles swans rabid skunks geese worms horses Edmontosorum (a variety of duck-billed dinosaurs) platypuses a very fat cat. Let us say we want to classify things according to being small and furry or being a duck-bill . We draw circles to display our classifications:

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Venn Diagram Now we will fill in, or "populate", the diagram. moles swans rabid skunks geese worms horses Edmontosorum (a variety of duck-billed dinosaurs) platypuses a very fat cat. Moles, rabid skunks, platypuses, and cat are all small and furry :

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Venn Diagram moles swans rabid skunks geese worms horses Edmontosorum (a variety of duck-billed dinosaurs) platypuses a very fat cat. Swans, geese, platypuses, and Edmontosorum are all duck-bills :

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Venn Diagram moles swans rabid skunks geese worms horses Edmontosorum (a variety of duck-billed dinosaurs) platypuses a very fat cat. Worms are small but not furry and horses are furry but not small, and neither is a duck-bill. However, they are animals; they fit inside our universe, but outside the circles.

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Venn Diagram What do you observe with the diagram? In other words, we really should have drawn the circles overlapped, like this:

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Venn Diagram Now when we populate the Venn diagram, we will only have to write "platypuses" once, in the overlap: moles swans rabid skunks geese worms horses Edmontosorum (a variety of duck-billed dinosaurs) platypuses a very fat cat. What do you observe with the circles?

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Venn Diagram and Set Notation The following examples should help you understand the notation, terminology, and concepts related to Venn diagrams and set notation. Let U = {1, 2, 3, 4} A  = {1, 2} and B = {2, 3} Then we have the following relationships, with pinkish shading marking the solution "regions" in the Venn diagrams:  

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Set Notation Pronunciation Meaning Venn diagram Answer A U B " A union B " everything that is in either of the sets   {1, 2, 3} " A intersect B " only the things that are in both of the sets   {2} A c or ~ A " A complement", or "not A " everything in the universe outside of   A   {3, 4} A ^ B or Venn Diagram and Set Notation Let U = {1, 2, 3, 4} A  = {1, 2} and B = {2, 3}

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Venn Diagram and Set Notation A – B " A minus B " , Or " A complement B " everything in A except for anything in its overlap with B   {1} ~( A U   B ) " not ( A union B )" everything outside A and B   {4} " not ( A intersect B )" everything outside of the overlap of A and   B   {1, 3, 4} Set Notation Pronunciation Meaning Venn diagram Answer ~( A ^ B ) or ~( A  B) Let U = {1, 2, 3, 4} A  = {1, 2} and B = {2, 3}

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Venn Diagram and Set Notation Describe the following sets: Wet clothes and dry clothes are separated A B Set A and B are disjoint sets All cheerleaders are girls Set D is a subset of Set C

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Venn Diagram and Set Notation Try this: Describe the following sets: Some boys and some girls are varsity players The intersection of Set E and Set F is composed of Varsity players

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Venn Diagram and Set Notation Observe the following Venn diagram and describe by providing the appropriate set operation.

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Venn Diagram and Set Notation Given the following set operations, provide an illustration using the Venn diagram. 1. 2. 3. 4.

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Venn Diagram Word Problems Venn diagram word problems generally give you two or three classifications and a bunch of numbers. You then have to use the given information to populate the diagram and figure out the remaining information. For instance… Out of forty students, 14 are taking English Composition and 29 are taking Chemistry. If five students are in both classes, how many students are in neither class? How many are in either class? What is the probability that a randomly-chosen student from this group is taking only the Chemistry class?

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Venn Diagram Word Problems Out of forty students, 14 are taking English Composition and 29 are taking Chemistry. If five students are in both classes, how many students are in neither class? How many are in either class? What is the probability that a randomly-chosen student from this group is taking only the Chemistry class? First, draw a universe for the forty students, with two overlapping circles labeled with the total in each:

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Venn Diagram Word Problems Out of forty students, 14 are taking English Composition and 29 are taking Chemistry. If five students are in both classes, how many students are in neither class? How many are in either class? What is the probability that a randomly-chosen student from this group is taking only the Chemistry class? Since five students are taking both classes, put "5" in the overlap:

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Venn Diagram Word Problems Out of forty students, 14 are taking English Composition and 29 are taking Chemistry. If five students are in both classes, how many students are in neither class? How many are in either class? What is the probability that a randomly-chosen student from this group is taking only the Chemistry class? We have now accounted for five of the 14 English students, leaving nine students taking English but not Chemistry, so we put "9" in the "English only" part of the "English" circle:

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Venn Diagram Word Problems Out of forty students, 14 are taking English Composition and 29 are taking Chemistry. If five students are in both classes, how many students are in neither class? How many are in either class? What is the probability that a randomly-chosen student from this group is taking only the Chemistry class? We have also accounted for five of the 29 Chemistry students, leaving 24 students taking Chemistry but not English, so we put "24" in the "Chemistry only" part of the "Chemistry" circle:

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Venn Diagram Word Problems Out of forty students, 14 are taking English Composition and 29 are taking Chemistry. If five students are in both classes, how many students are in neither class? How many are in either class? What is the probability that a randomly-chosen student from this group is taking only the Chemistry class? This tells us that a total of 9 + 5 + 24 = 38 students are in either English or Chemistry (or both). This leaves two students unaccounted for, so they must be the ones taking neither class.

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Venn Diagram Word Problems Out of forty students, 14 are taking English Composition and 29 are taking Chemistry. If five students are in both classes, how many students are in neither class? How many are in either class? What is the probability that a randomly-chosen student from this group is taking only the Chemistry class? *Two students are taking neither class. *There are 38 students in at least one of the classes. *There is a 24/40 = 0.6 = 60% probability that a randomly-chosen student in this group is taking Chemistry but not English. From this populated Venn diagram, I can get the answers to the questions.

The Venn diagram is also used to analyse different relationship that exists between two or more objects. :

The Venn diagram is also used to analyse different relationship that exists between two or more objects. Into swimming but not basketball? 2. Into both swimming and basketball? 3. Into either swimming or basketball? 4. Neither into swimming nor basketball? Venn Diagram Word Problems The Venn diagram below shows student involvement in two sports. If 100 students were surveyed, how many students were:

140 students were surveyed. The Venn diagram shows the number of students who enjoy singing and/or dancing. How many students enjoy::

140 students were surveyed. The Venn diagram shows the number of students who enjoy singing and/or dancing. How many students enjoy: Dancing but not singing? 2. Singing or dancing? 3. Singing and dancing? 4. Neither singing nor dancing? Venn Diagram Word Problems

Fifty people are asked about the pets they keep at home. The Venn diagram shows the result. Let D = { people who have dogs} F = { people who have fish } C = { people who have cats }:

Fifty people are asked about the pets they keep at home. The Venn diagram shows the result. Let D = { people who have dogs} F = { people who have fish } C = { people who have cats } Venn Diagram Word Problems How many people have a. Dogs? b. Dogs and fish? c. Dogs or cats? d. Fish and cats but not dogs? e. Dogs or fish but not cats? f. All three? g. Neither one of the three?

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Venn Diagram Word Problems 100 people were surveyed. Suppose: B = { people who like beaches } M = { people who like mountains } C = { people who like big cities } How many people like staying in: 1. big cities only? 2. mountains or beaches? 3. mountains and beaches but not big cities? 4. cities or beaches but not mountains? 5. all three? 6. Neither of the three?

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In a high school, students know at least one of the dialects, Tagalog and Cebuano. 201 students know Tagalog, 114 students know Cebuano and 100 of them know both Tagalog and Cebuano. How many students are there in the school?   Venn Diagram Word Problems

Solution:

Solution Cebuano Tagalog 100 101 14

Problem # 2:

Problem # 2 In a class of 15 boys, there are 10 who play basketball and 8 play chess. How many play both of the games? How many play basketball only? How many play chess only?

Solution:

Solution 10 8 Total Numbers of boys = 15

Solution:

Solution 7 3 5

Problem # 3:

Problem # 3 In a class, 40 can speak either English or Tagalog or both. If 25 students can speak English and 20 can speak both; find the number of those who can speak Tagalog only. 15 students

Solution:

Solution

Problem # 4:

Problem # 4 In a group of 79 persons, 29 drink Coke but not Pepsi and 42 drink Coke. How many drink Pepsi but not Coke? How many drink both? The answers are: a. 37 b. 13

Problem # 5:

Problem # 5 A class has 175 students. The following is the description of students studying one or more of the following subjects in this class. Mathematics = 100 Physics = 70 Chemistry = 46 Mathematics and Physics = 30 Mathematics and Chemistry = 28 Physics and Chemistry = 23 All three = 18 Questions: How many students are enrolled in Mathematics Alone? Physics alone? Chemistry Alone? 60 35 13

Historical Note:

Historical Note John Venn (4 August 1834 – 4 April 1923), was a British logician and philosopher. He is famous for introducing the Venn diagram, which is used in many fields, including set theory, probability, logic, statistics, and computer science.

Summing up!!!:

Summing up!!! Can you share to the class some new terms you learned today? Which definition struck you the most? If you can share a topic that you learned today to a friend, what would it be? Why?

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