https://file.aiquickdraw.com/mathbot/file/17261912279584zlct4my.jpg

Let's analyze the Venn diagram and the questions in your quiz about sets.

### Venn Diagram:
- Three sets: \( A \) (Blue), \( B \) (Green), and \( C \) (Red).
- The cardinality (number of elements) of each intersection is provided within the Venn diagram.

From the Venn diagram, we can see the following cardinalities:
- \( n(A \cap B \cap C) = 3 \) (the intersection of all three sets).
- \( n(A \cap B) = 7 \) (including the \(3\) that belongs to all three sets, so \( 7 - 3 = 4 \) in \( A \cap B \) but not in \( C \)).
- \( n(A \cap C) = 6 \) (the region where \( A \) and \( C \) overlap, but not \( B \), this includes the central intersection as well).
- \( n(B \cap C) = 8 \) (including the 3 in the middle, leaving 5 elements that are in both \( B \) and \( C \), but not \( A \)).

### Now, let's address the specific questions in the quiz:

1. **\( n(A) \)**:
   To find \( n(A) \), we sum all regions that belong to set \( A \):
   - \( 8 \) (only in \( A \)),
   - \( 7 \) (overlap between \( A \) and \( B \)),
   - \( 6 \) (overlap between \( A \) and \( C \)),
   - \( 3 \) (overlap between \( A \), \( B \), and \( C \)).
   
   So, \( n(A) = 8 + 4 + 3 + 3 = 20 \).

2. **\( n(A \cap C) \)**:
   This is the intersection of sets \( A \) and \( C \), including the overlap with \( B \):
   - \( 6 \) (only \( A \cap C \)),
   - \( 3 \) (intersection of all three sets).
   
   So, \( n(A \cap C) = 6 + 3 = 9 \).

3. **\( n(A \cap B \cap C^c) \)**:
   This asks for the intersection of \( A \) and \( B \), but excluding \( C \) (the complement of \( C \)).
   From the diagram:
   - The region that is in \( A \cap B \) but not in \( C \) is the area with \( 4 \) elements (since the middle \(3\) elements overlap with \( C \)).

   So, \( n(A \cap B \cap C^c) = 4 \).

Would you like more details on this explanation?

### Related Questions:
1. What is \( n(B) \) based on the Venn diagram?
2. Can you calculate \( n(C) \)?
3. What is \( n(A \cup B) \)?
4. What is the total number of elements in all sets combined?
5. How does the inclusion-exclusion principle work for this Venn diagram?

### Tip:
When working with Venn diagrams and set cardinalities, always break the intersections down carefully, especially for overlapping regions, to avoid double-counting elements.

The Venn diagram here shows the cardinality of each set. Use this to find the cardinality of each given set.

Solve a set theory problem using a Venn diagram to find the cardinality of given sets. This problem involves finding n(A), n(A ∩ C), and n(A ∩ B ∩ C^c) using set intersections and the inclusion-exclusion principle. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation