### 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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```  ____      ______    _  __   _    _     ______
|  _ \\   /_   _//  | |/ // | || | ||  /_   _//
| |_| ||   -| ||-   | ' //  | || | ||    | ||
| .  //    _| ||_   | . \\  | \\_/ ||   _| ||
|_|\_\\   /_____//  |_|\_\\  \____//   /__//
`-` --`   `-----`   `-` --`   `---`    `--`

```
• 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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``` __   __   _    _     ______    ______   _    _
\ \\/ // | || | ||  /_____//  /_   _// | |  | ||
\   //  | || | ||  `____ `    -| ||-  | |/\| ||
/ . \\  | \\_/ ||  /___//     _| ||_  |  /\  ||
/_//\_\\  \____//   `__ `     /_____// |_// \_||
`-`  --`   `---`    /_//      `-----`  `-`   `-`
`-`
```
• 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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``` __   _     _____                ___     _____
| || | ||  |  ___||     ___     / _ \\  |  __ \\
| '--' ||  | ||__      /   ||  | / \ || | |  \ ||
| .--. ||  | ||__     | [] ||  | \_/ || | |__/ ||
|_|| |_||  |_____||    \__ ||   \___//  |_____//
`-`  `-`   `-----`      -|_||   `---`    -----`
`-`
```
• 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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```   _____    ______   _____       ___      _____
/ ___//  /_   _// |  __ \\    / _ \\   / ____||
\___ \\   -| ||-  | |  \ ||  / //\ \\ / //---`'
/    //   _| ||_  | |__/ || |  ___  ||\ \\___
/____//   /_____// |_____//  |_||  |_|| \_____||
`-----`    `-----`   -----`   `-`   `-`   `----`

```
• 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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``` _    _    __   __  __    __     ___     _    _
| || | ||  \ \\/ // \ \\ / //   / _ \\  | \  / ||
| || | ||   \ ` //   \ \/ //   | / \ || |  \/  ||
| \\_/ ||    | ||     \  //    | \_/ || | .  . ||
\____//     |_||      \//      \___//  |_|\/|_||
`---`      `-`'       `       `---`   `-`  `-`

```
• 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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``` ______    _    _    _    _    __   __   ______
|      \\ | || | || | |  | ||  \ \\/ // |      \\
|  --  // | || | || | |/\| ||   \ ` //  |  --  //
|  --  \\ | \\_/ || |  /\  ||    | ||   |  --  \\
|______//  \____//  |_// \_||    |_||   |______//
`------`    `---`   `-`   `-`    `-`'   `------`

```
• 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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```__    __    ______    _____    _    _    __   __
\ \\ / //  /_   _//  |__  //  | || | ||  \ \\/ //
\ \/ //    -| ||-     / //   | || | ||   \   //
\  //     _| ||_    / //__  | \\_/ ||   / . \\
\//     /_____//  /_____||  \____//   /_//\_\\
`      `-----`   `-----`    `---`    `-`  --`

```
• 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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``` _    _     _____    __   _      ___      ______