Venn Diagrams

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  Crossing the River with Dogs

  Problem Solving for College Students

  • Ken Johnson, Ted Herr, Judy Kysh(Authors)
  • 2018(Publication Date)
  • Wiley

    (Publisher)

  D R A W V E N N D I A G R A M S 323 12 12 Draw Venn Diagrams T his specialized strategy is a method of sorting out the elements of overlapping categories. When you recycle, you separate recyclable materials from other trash, then further sort these materials into subsets such as aluminum, glass, and paper. Drawing Venn Diagrams is a helpful strategy for solving problems involving objects that can be sorted into different categories. 324 C H A P T E R 1 2 athematics students often use Venn Diagrams to categorize things because Venn Diagrams can clearly show relationships among different categories. As a problem-solving strategy, Venn Diagrams fall into the major theme of Spatial Organization. They organize information in a particular way that otherwise could be hard to see. When you draw Venn Diagrams, you can use loops, closed circles, or rectangles. Two loops can intersect or be entirely disjoint, or one can be completely inside the other. Each pair of loops represents a different type of relationship between two categories. In certain types of problems, you will encounter the word all, some, or no. For example, the word all could be used in a statement such as “All roses are flowers.” Using a Venn diagram can help you correctly interpret the meaning of these three important words and of other words in a problem. Several words of caution: When we talk about correctly interpreting the meanings of words in Venn diagram problems, we are talking about the principal, common understanding of the words. For example, you may know a woman named Rose, but the name Rose is not the principal, common understanding of the word rose. Don’t waste time looking for obscure meanings or interpretations that will render all the information in a problem false. Consider the statement “All roses are flowers.” Here is a Venn interpretation of this statement. The diagram shows the word Flowers. Anything inside the rectangle is considered to be a flower.

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  Probability For Dummies

  • Deborah J. Rumsey(Author)
  • 2018(Publication Date)
  • For Dummies

    (Publisher)

  The S in the lower-right corner of each box indicates that the entire set is the sample space S. You can omit this notation unless you reduce the sample space; for example, if you look only at the group of females in a class, you can write F in that space to indicate your new sample space.

  Utilizing Venn Diagrams to find probabilities beyond those given

  The most important use of a Venn diagram is to help you find probabilities beyond those given in the problem. You know the drill; you’re given certain pieces of information in a problem, and you have to answer tons of questions with the pieces. Sometimes it seems like you have to take straw and turn it into gold, right? With a Venn diagram, you can organize the information you have, and with the rules of sets and probability, the diagram helps you organize, identify, and figure out other probabilities that you need to find.

  Before you start to work out your solution to the problem, set up and fill out your Venn diagram completely first. That’s the key to success.

  Using Venn Diagrams to organize and visualize relationships

  Venn Diagrams help you organize and account for all the possible sets and subsets that occur in a probability scenario. Each piece of the diagram has meaning and a probability. After you account for all the pieces in terms of their probabilities, you can solve many different types of problems. In Figure 3-2 , you see a Venn diagram representing set A and its complement, Ac (the shaded area represents Ac ; the complement of a set A is everything in the sample space that isn’t included in set A [see Chapter 2 ]).

  Figure 3-2: Sets A and Ac , represented by a Venn diagram.

  Venn Diagrams also help you to visualize important relationships that can exist between two events. In Figure 3-3 , I show two sets, A and B, which represent two events, A and B, that can intersect, and I identify and label each part of each piece of the diagram in terms of set notation. The set represents the set of all outcomes in S that appear in both A and B. The set represents all outcomes in S that appear in A but not in B. The set represents all outcomes in S that appear in B but not in A.

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  Mathematics

  A Practical Odyssey

  • David Johnson, , Thomas Mowry, , David Johnson, Thomas Mowry(Authors)
  • 2015(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  Venn Diagrams are particularly useful in survey analysis. n Surveys Surveys are often used to divide people or objects into categories. Because the cate-gories sometimes overlap, people can fall into more than one category. Venn dia-grams and the rules for cardinal numbers can help researchers organize the data. ANALYZING THE RESULTS OF A SURVEY: TWO SETS Has the ad-vent of the DVD affected attendance at movie theaters? To study this question, Professor Redrum’s film class conducted a survey of people’s movie-watching habits. He had his students ask hundreds of people between the ages of sixteen and forty-five to check the appropriate box or boxes on the following form: I watched a movie in a theater during the past month. I watched a movie on a DVD during the past month. After the professor had collected the forms and tabulated the results, he told the class that 388 people had checked the theater box, 495 had checked the DVD box, 281 had checked both boxes, and 98 of the forms were blank. Giving the class only this information, Professor Redrum posed the following three questions. a. What percent of the people surveyed watched a movie in a theater or on a DVD during the past month? b. What percent of the people surveyed watched a movie in a theater only? c. What percent of the people surveyed watched a movie on a DVD only? EXAMPLE 1 a. To calculate the desired percentages, we must determine n ( U ), the total number of people surveyed. This can be accomplished by drawing a Venn diagram. Because the survey divides people into two categories (those who watched a movie in a theater and those who watched a movie on a DVD), we need to define two sets. Let T 5 {people u the person watched a movie in a theater} D 5 {people u the person watched a movie on a DVD} Now translate the given survey information into the symbols for the sets and attach their given cardinal numbers: n ( T ) 5 388, n ( D ) 5 495, and n ( T ¨ D ) 5 281.

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  Mathematical Excursions

  • Richard Aufmann, Joanne Lockwood, Richard Nation, Daniel K. Clegg(Authors)
  • 2017(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  ( Source: http://www.combinatorics.org/Surveys/ ds5/VennWhatEJC.html) 1 Anthony W. F. Edwards, “Venn Diagrams for many sets,” New Scientist, 7 January 1989, pp. 51–56. Difference of Sets Another operation that can be defined on sets A and B is the difference of the sets, denoted by A - B. Here is a formal definition of the differ- ence of sets A and B. A - B = h x u x [ A and x ∉ Bj Thus A - B is the set of elements that belong to A but not to B. For instance, let A = h 1, 2, 3, 7, 8j and B = h 2, 7, 11j. Then A - B = h 1, 3, 8j. ■ In Exercises 71 to 76, determine each difference, given that U = h 1, 2, 3, 4, 5, 6, 7, 8, 9j, A = h 2, 4, 6, 8j, and B = h 2, 3, 8, 9j. 71. B - A 72. A - B 73. A - B 74. B - A 75. A - B 76. A - B 77. John Venn Write a few paragraphs about the life of John Venn and his work in the area of mathematics. The following Venn diagram illustrates that four sets can partition the universal set into 16 different regions. B C A D U EXTENSIONS Copyright 2018 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 80 CHAPTER 2 | Sets SECTION 2.4 Applications of Sets Surveys: An Application of Sets Counting problems occur in many areas of applied mathematics. To solve these counting problems, we often make use of a Venn diagram and the inclusion-exclusion principle, which will be presented in this section. EXAMPLE 1 A Survey of Preferences A movie company is making plans for future movies it wishes to produce. The company has done a random survey of 1000 people. The results of the survey are shown below. 695 people like action adventures.

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  Mathematics for Elementary School Teachers

  • Tom Bassarear, Meg Moss(Authors)
  • 2015(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  What percent of the people in the area only heard the ad on radio or only saw the ad on TV? 20. Refer to the discussion about the book Mathematics in Science and Society in the last subsection. a. What would be the focus of the authors’ discussion following the Venn diagram below? Describe your answer in everyday English. Lab approaches Concrete materials Small groups Unless otherwise noted, all content on this page is © Cengage Learning Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Section 2.2 Numeration 53 b. The following Venn diagram appeared in the same book. Lab approaches Concrete materials Small groups 21. Make up a situation for which the following Venn diagram is appropriate. 22. Consider the following two subsets of U , the set of all people: I 5 the set of intelligent people S 5 the set of successful people a. By yourself, answer the following question: How would you represent these two subsets with a Venn diagram? b. Discuss your responses with other classmates. What did you learn from this activity? How did the Venn diagram contrib-ute to this learning? 23. Many Internet math activities are on the site www.shodor .org/interactivate/activities/. There are three there with Venn Diagrams. When you go to this address, type in “Venn” in the search tool. The three are called “Triple Venn Diagram Shape Sorter,” “Venn Diagram Shape Sorter,” and “Venn Diagrams.” Explore each of these. There is a “help” tab at the top of each activity if you need help figuring them out.

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