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3.1 Sets & Set Diagrams

3.1 Sets & Set Diagrams
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  • 1. College Testing

    In a high school graduating class, 128 students took the SAT, 100 students took the ACT, and 98 students took both the SAT and the ACT. Draw a set diagram that shows the information. (See Example 1.)

    • Worked-Out Solution

      In your set diagram, begin by drawing two overlapping circles to represent the students who took each test.

      You are given that 98 students took both the SAT and the ACT. To find the number of students who took only the SAT, subtract.

      To find the number of students who took only the ACT, subtract.

      Now, you can complete the set diagram with the number of students listed in each of the three regions.

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     _    _    __   __     _____     ___    __    __  
    | \  / ||  \ \\/ //   / ___//   / _ \\  \ \\ / // 
    |  \/  ||   \ ` //    \___ \\  | / \ ||  \ \/ //  
    | .  . ||    | ||     /    //  | \_/ ||   \  //   
    |_|\/|_||    |_||    /____//    \___//     \//    
    `-`  `-`     `-`'   `-----`     `---`       `     
                                                      
    
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  • 2. College Testing

    In a high school graduating class, 128 students took the SAT, 100 students took the ACT, and 98 students took both the SAT and the ACT. How many students took the SAT or the ACT? (See Example 1.)

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     _    _    __   __   _    _      ___      _____   
    | || | ||  \ \\/ // | \  / ||   / _ \\   |__  //  
    | || | ||   \ ` //  |  \/  ||  | / \ ||    / //   
    | \\_/ ||    | ||   | .  . ||  | \_/ ||   / //__  
     \____//     |_||   |_|\/|_||   \___//   /_____|| 
      `---`      `-`'   `-`  `-`    `---`    `-----`  
                                                      
    
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  • 3. College Programs

    The set diagram shows characteristics of colleges that a high school senior is considering. How many of the colleges offer an engineering program? (See Example 2.)

    • Worked-Out Solution

      To find the number of colleges that offer an engineering program, add the numbers in the 8 regions that make up that set.

      So, 17 of the colleges offer an engineering program.

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     ______      ___     _    _    _    _     _  __  
    |      \\   / _ \\  | |  | || | || | ||  | |/ // 
    |  --  //  / //\ \\ | |/\| || | || | ||  | ' //  
    |  --  \\ |  ___  |||  /\  || | \\_/ ||  | . \\  
    |______// |_||  |_|||_// \_||  \____//   |_|\_\\ 
    `------`  `-`   `-` `-`   `-`   `---`    `-` --` 
                                                     
    
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  • 4. College Programs

    The set diagram shows characteristics of colleges that a high school senior is considering. How many of the colleges offer sports scholarships or an engineering program? (See Example 2.)

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     _____     __   __   _    _    __   __            
    |  __ \\   \ \\/ // | |  | ||  \ \\/ //     ___   
    | |  \ ||   \ ` //  | |/\| ||   \ ` //     /   || 
    | |__/ ||    | ||   |  /\  ||    | ||     | [] || 
    |_____//     |_||   |_// \_||    |_||      \__ || 
     -----`      `-`'   `-`   `-`    `-`'       -|_|| 
                                                 `-`  
    
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  • 5. College Programs

    The set diagram shows characteristics of colleges that a high school senior is considering. How many of the colleges offer sports scholarships and have an enrollment less than 2000? (See Example 2.)

    • Worked-Out Solution

      To find the number of colleges that offer sports scholarships and have an enrollment of less than 2000, add the number in the 4 regions that represent this overlap.

      So, 7 of the colleges offer sports scholarships and have an enrollment of less than 2000.

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      ______     ___      ______   _    _     ____    
     /_   _//   / _ \\   /_____// | || | ||  |  _ \\  
     `-| |,-   / //\ \\  `____ `  | || | ||  | |_| || 
       | ||   |  ___  || /___//   | \\_/ ||  | .  //  
       |_||   |_||  |_|| `__ `     \____//   |_|\_\\  
       `-`'   `-`   `-`  /_//       `---`    `-` --`  
                         `-`                          
    
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  • 6. College Programs

    The set diagram shows characteristics of colleges that a high school senior is considering. How many of the colleges have tuition less than $20,000 per year, offer an engineering program, and offer sports scholarships? (See Example 2.)

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     ______     _____    _    _      ___      ______  
    |      \\  |  ___|| | |  | ||   / _ \\   /_   _// 
    |  --  //  | ||__   | |/\| ||  / //\ \\  `-| |,-  
    |  --  \\  | ||__   |  /\  || |  ___  ||   | ||   
    |______//  |_____|| |_// \_|| |_||  |_||   |_||   
    `------`   `-----`  `-`   `-` `-`   `-`    `-`'   
                                                      
    
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  • 7. Enrollment and Tuition

    The set diagram shows characteristics of colleges that a high school senior is considering. Use the set diagram to determine whether colleges with an enrollment less than 2000 are likely to have tuitions less than $20,000 per year. Explain why the answer is reasonable in real life. (See Example 2.)

    • Worked-Out Solution

      There are

      colleges that have enrollments of less than 2000. Of these,

      have a tuition that is less than $20,000 a year. This does make some sense in real life because small 4-year colleges need to charge a substantial amount in tuition to meet their expenses. Note that 2000 students at $20,000 each for tuition is a total annual budget of $40 million. This might sound like a lot, but for running a college, it is a tight budget.

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     __   _      ___      _____      ___      _____   
    | || | ||   / _ \\   /  ___||   / _ \\   |__  //  
    | '--' ||  | / \ || | // __    | / \ ||    / //   
    | .--. ||  | \_/ || | \\_\ ||  | \_/ ||   / //__  
    |_|| |_||   \___//   \____//    \___//   /_____|| 
    `-`  `-`    `---`     `---`     `---`    `-----`  
                                                      
    
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  • 8. Researching Colleges

    List three characteristics of a college that are important to you. Research 20 colleges to determine whether each college meets your desired characteristics. Create a set diagram of the results.

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      ______     ___     _    _      ___     __   _   
     /_   _//   / _ \\  | \  / ||   / _ \\  | || | || 
       | ||    / //\ \\ |  \/  ||  / //\ \\ | '--' || 
      _| ||   |  ___  ||| .  . || |  ___  ||| .--. || 
     /__//    |_||  |_|||_|\/|_|| |_||  |_|||_|| |_|| 
     `--`     `-`   `-` `-`  `-`  `-`   `-` `-`  `-`  
                                                      
    
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