https://file.aiquickdraw.com/mathbot/file/1725537533942nhkspnze.jpg

The image contains a Venn diagram with three sets, where:

- \( P \) represents the set of players who play football,
- \( Q \) represents the set of players who play tennis,
- \( R \) represents the set of players who play hockey.

The problem involves determining the number of players who are part of each set, based on the diagram provided. From the Venn diagram:

- 5 players are only in \( P \),
- 3 players are in both \( P \) and \( Q \),
- 2 players are in all three sets \( P \), \( Q \), and \( R \),
- 4 players are in both \( P \) and \( R \),
- 8 players are only in \( R \),
- 7 players are only in \( Q \).

The task involves solving parts related to set theory. To proceed, let's break down the steps:

### Questions:
1. How many players are in \( P \cup Q \cup R \)? This is the total number of players in at least one of the sports.
2. How many players are in exactly two sports?
3. How many players are in only one sport?
4. What is the number of players who are in \( P \cap Q \cap R \) (i.e., all three sports)?
5. How many players are in neither \( P \), \( Q \), nor \( R \)?

Would you like to proceed with these calculations or need any specific clarifications? 

Also, here are some relative questions:
1. What formula is used to find the union of three sets?
2. How does the intersection of two sets affect the union of three sets?
3. What is the principle of inclusion-exclusion in set theory?
4. How do we interpret elements that belong to two sets but not the third?
5. How can Venn diagrams simplify set operations?

**Tip:** When using Venn diagrams, always check for overlapping areas to ensure all subsets are counted correctly.

The image shows a Venn diagram with three sets representing players who play football, tennis, and hockey. The goal is to determine the number of players involved in different combinations of these sports based on the provided numbers in the diagram.

Explore how to solve set theory problems involving football, tennis, and hockey using a Venn diagram. Learn about the principle of inclusion-exclusion and how to find unions and intersections of sets in this problem, suitable for students in Grades 9-11.

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