Of the 90 students involved in the karate club:

47

students are working on a belts

35

students are taking thai boxing

31

students are taking grappling

19	students are taking both thai boxing and grappling
13	students are working on a belt and taking grappling
14	students are working on a belt and taking thai boxing
11	students are taking all three classes
Draw a properly labelled Venn Diagram that corresponds with the information above. Make sure to show the work of how you determine the numbers for each category.

(For this question you can draw the diagram on paper and upload a photo or create it with technology and upload an image or attachment file below)

To create the Venn Diagram, we need to follow these steps:

1. **Identify the Sets:**
   - Let \(A\) be the set of students working on a belt.
   - Let \(B\) be the set of students taking Thai boxing.
   - Let \(C\) be the set of students taking grappling.

2. **Given Values:**
   - \( |A| = 47 \)
   - \( |B| = 35 \)
   - \( |C| = 31 \)
   - \( |B \cap C| = 19 \)
   - \( |A \cap C| = 13 \)
   - \( |A \cap B| = 14 \)
   - \( |A \cap B \cap C| = 11 \)

3. **Find the Values for Each Region:**
   - Students taking all three classes: \( |A \cap B \cap C| = 11 \)
   - Students taking both Thai boxing and grappling but not working on a belt: \( |B \cap C| - |A \cap B \cap C| = 19 - 11 = 8 \)
   - Students working on a belt and taking grappling but not taking Thai boxing: \( |A \cap C| - |A \cap B \cap C| = 13 - 11 = 2 \)
   - Students working on a belt and taking Thai boxing but not taking grappling: \( |A \cap B| - |A \cap B \cap C| = 14 - 11 = 3 \)
   - Students only working on a belt: \( |A| - (only belt & Thai boxing + only belt & grappling + all three) = 47 - (3 + 2 + 11) = 31 \)
   - Students only taking Thai boxing: \( |B| - (only Thai boxing & grappling + only belt & Thai boxing + all three) = 35 - (8 + 3 + 11) = 13 \)
   - Students only taking grappling: \( |C| - (only belt & grappling + only Thai boxing & grappling + all three) = 31 - (2 + 8 + 11) = 10 \)

4. **Calculate Total Overlap:**
   - Sum of all regions inside the Venn Diagram: \( 31 + 13 + 10 + 3 + 2 + 8 + 11 = 78 \)
   - Students involved in at least one activity: \( 90 \)

5. **Students not involved in any activity:**
   - \( 90 - 78 = 12 \)

Now, let's draw the Venn Diagram based on these values:

1. **Regions and their values:**
   - Only A (belt): \( 31 \)
   - Only B (Thai boxing): \( 13 \)
   - Only C (grappling): \( 10 \)
   - A and B only: \( 3 \)
   - A and C only: \( 2 \)
   - B and C only: \( 8 \)
   - All three (A, B, and C): \( 11 \)
   - Outside all sets: \( 12 \)

![Venn Diagram](https://www.researchgate.net/profile/Sitharam-Jayasankar/publication/226862375/figure/fig6/AS:341574270873605@1458424389291/A-three-set-Venn-diagram-with-regions-labelled-according-to-their-constituent-subsets.png)

To help visualize this, here is a verbal description:

- Draw three overlapping circles.
- Label the left circle \( A \), the top circle \( B \), and the right circle \( C \).
- Place the following numbers in their appropriate regions:

  - **31** in \( A \) only.
  - **13** in \( B \) only.
  - **10** in \( C \) only.
  - **3** in \( A \cap B \) (not including \( C \)).
  - **2** in \( A \cap C \) (not including \( B \)).
  - **8** in \( B \cap C \) (not including \( A \)).
  - **11** in \( A \cap B \cap C \) (all three).

- Place **12** outside all circles to represent students not involved in any activity.

Would you like more details or have any questions?

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Here are five questions you might ask:

1. How do you determine the number of students involved in at least one activity?
2. What method is used to find the number of students not involved in any activity?
3. Can you explain how to derive the individual values for each region in the Venn Diagram?
4. What is the importance of checking the total sum of all regions within the Venn Diagram?
5. How can Venn Diagrams be useful in solving real-life problems?

**Tip:** When dealing with Venn Diagrams, always check if the sum of all regions matches the total number of elements in the universal set to ensure accuracy.

Learn how to solve set theory problems using Venn diagrams with a detailed example and solution. This problem involves determining student involvement in karate club activities such as belt work, Thai boxing, and grappling. Suitable for students in Grades 7-9.

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