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3.2 Statements & Negations

3.2 Statements & Negations
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  • 15. New Year's Resolutions

    Use a set diagram to visualize the negation of the New Year's resolution. (See Example 5 and Example 6.)

    I will save $100 each month and go on a vacation overseas.

    • Worked-Out Solution

      So, the resolution can be broken in three ways.

      • Miss saving $100 for at least one month.
      • Fail to take a vacation overseas.
      • Miss a month of saving and fail to take a vacation overseas.
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     __   _     ______    ____     __   __    ______  
    | || | ||  /_   _//  |  _ \\   \ \\/ //  /_   _// 
    | '--' ||   -| ||-   | |_| ||   \ ` //   `-| |,-  
    | .--. ||   _| ||_   | .  //     | ||      | ||   
    |_|| |_||  /_____//  |_|\_\\     |_||      |_||   
    `-`  `-`   `-----`   `-` --`     `-`'      `-`'   
                                                      
    
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  • 16. New Year's Resolutions

    Use a set diagram to visualize the negation of the New Year's resolution. (See Example 5 and Example 6.)

    I will exercise 4 times each week and lose 10 pounds.

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      ______   __   __  __    __     ___    __    __  
     /_   _//  \ \\/ // \ \\ / //   / _ \\  \ \\ / // 
       | ||     \ ` //   \ \/ //   / //\ \\  \ \/ //  
      _| ||      | ||     \  //   |  ___  ||  \  //   
     /__//       |_||      \//    |_||  |_||   \//    
     `--`        `-`'       `     `-`   `-`     `     
                                                      
    
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  • 17. New Year's Resolutions

    Use a set diagram to visualize the negation of the New Year's resolution. (See Example 5 and Example 6.)

    I will volunteer at a homeless shelter or a soup kitchen.

    • Worked-Out Solution

      So, you will break your New Year's resolution if you do not volunteer at the homeless shelter and you do not volunteer at the soup kitchen.

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      ______   __   __   _    _      ___     ______   
     /_   _//  \ \\/ // | || | ||   / _ \\  |      \\ 
     `-| |,-    \ ` //  | || | ||  | / \ || |  --  // 
       | ||      | ||   | \\_/ ||  | \_/ || |  --  \\ 
       |_||      |_||    \____//    \___//  |______// 
       `-`'      `-`'     `---`     `---`   `------`  
                                                      
    
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  • 18. New Year's Resolutions

    Use a set diagram to visualize the negation of the New Year's resolution. (See Example 5 and Example 6.)

    I will stop biting my fingernails or cracking my knuckles.

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     ______    __   __   _____     _    _     _____   
    |      \\  \ \\/ // |  __ \\  | || | ||  /  ___|| 
    |  --  //   \ ` //  | |  \ || | || | || | // __   
    |  --  \\    | ||   | |__/ || | \\_/ || | \\_\ || 
    |______//    |_||   |_____//   \____//   \____//  
    `------`     `-`'    -----`     `---`     `---`   
                                                      
    
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  • 19. Employee Handbook

    Use a set diagram to visualize the negation of the statement about holidays in a company handbook. (See Example 5 and Example 6.)

    Employees will not be paid for vacations, holidays, and sick leave.

    • Worked-Out Solution

      Whatever job this is, it is apparently one in which employees receive no paid days off. There are many ways to use set diagrams to visualize the statement. Here is one way.

      For the statement to not be true, there must be at least one vacation day or one holiday or one sick day in which at least one employee was paid.

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      ______    _____     _  __    _____   __    __  
     /_   _//  |  ___||  | |/ //  |  ___|| \ \\ / // 
       | ||    | ||__    | ' //   | ||__    \ \/ //  
      _| ||    | ||__    | . \\   | ||__     \  //   
     /__//     |_____||  |_|\_\\  |_____||    \//    
     `--`      `-----`   `-` --`  `-----`      `     
                                                     
    
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  • 20. Employee Handbook

    Use a set diagram to visualize the negation of the statement about holidays in a company handbook. (See Example 5 and Example 6.)

    Employees do not have to work the Friday before Easter, September 11, and holidays for federal employees.

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     ______      ___    __    __     ___     _    _   
    |      \\   / _ \\  \ \\ / //   / _ \\  | || | || 
    |  --  //  | / \ ||  \ \/ //   / //\ \\ | || | || 
    |  --  \\  | \_/ ||   \  //   |  ___  ||| \\_/ || 
    |______//   \___//     \//    |_||  |_|| \____//  
    `------`    `---`       `     `-`   `-`   `---`   
                                                      
    
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  • 21. Employee Handbook

    Use a set diagram to visualize the negation of the statement about holidays in a company handbook. (See Example 5 and Example 6.)

    Employees will be invited to a company picnic on Memorial Day, Independence Day, or Labor Day.

    • Worked-Out Solution

      The original statement is an "or" statement. This implies that the negation will be an "and" statement.

      Negation: At least one employee will not be invited to a company picnic on Memorial Day, Independence Day, and Labor Day.

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      ______    ______    _  _     ______   __   __  
     /_   _//  /_   _//  | \| ||  /_   _//  \ \\/ // 
       | ||     -| ||-   |  ' ||   -| ||-    \   //  
      _| ||     _| ||_   | .  ||   _| ||_    / . \\  
     /__//     /_____//  |_|\_||  /_____//  /_//\_\\ 
     `--`      `-----`   `-` -`   `-----`   `-`  --` 
                                                     
    
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  • 22. Compensation

    The following statement is in a company's employee handbook.

    Employees will receive a cost-of-living adjustment each year or a New Year's Day bonus.

    1. Use a set diagram to visualize the negation of the statement. (See Example 6.)
    2. The company changes the word or to and. How does this change the negation of the statement? (See Example 5.)

       

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    __    __    _____     _____    __   __   _____    
    \ \\ / //  |  ___||  /  ___||  \ \\/ // |  __ \\  
     \ \/ //   | ||__   | // __     \ ` //  | |  \ || 
      \  //    | ||__   | \\_\ ||    | ||   | |__/ || 
       \//     |_____||  \____//     |_||   |_____//  
        `      `-----`    `---`      `-`'    -----`   
                                                      
    
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