https://file.aiquickdraw.com/mathbot/file/172799869275072mgp95g.jpg

Let's analyze the Venn diagram and the set notations in the image you provided. We will calculate the cardinalities step by step.

### Venn Diagram Breakdown:
The Venn diagram consists of three sets \( A \), \( B \), and \( C \). Here are the values for each region:

- Only \( A \): 11
- Only \( B \): 7
- Only \( C \): 9
- \( A \cap B \): 2
- \( A \cap C \): 6
- \( B \cap C \): 5
- \( A \cap B \cap C \): 3

Now, let's solve the questions:

### (a) \( n(A^c) = 22 \)
This value is already provided. It represents the number of elements outside of set \( A \).

### (b) \( n(A \cup B) \)
To calculate the union of sets \( A \) and \( B \), we sum all the values in regions where \( A \), \( B \), or both sets are present. That includes:

- Only \( A \): 11
- Only \( B \): 7
- \( A \cap B \): 2
- \( A \cap C \cap B \): 3 (since it's part of both \( A \) and \( B \))

So:
\[
n(A \cup B) = 11 + 7 + 2 + 3 = 23
\]

### (c) \( n(A \cap C^c) \)
The complement of \( C \) is all the regions outside of \( C \). So, we consider elements in \( A \) that do not intersect with \( C \), which includes:

- Only \( A \): 11
- \( A \cap B \): 2

So:
\[
n(A \cap C^c) = 11 + 2 = 13
\]

### (d) \( n(A \cap B \cap C) \)
This is given directly by the diagram:
\[
n(A \cap B \cap C) = 3
\]

### (e) \( n(A \cup B \cup C) \)
For the union of all three sets \( A \), \( B \), and \( C \), we sum all regions in the Venn diagram:

- Only \( A \): 11
- Only \( B \): 7
- Only \( C \): 9
- \( A \cap B \): 2
- \( A \cap C \): 6
- \( B \cap C \): 5
- \( A \cap B \cap C \): 3

So:
\[
n(A \cup B \cup C) = 11 + 7 + 9 + 2 + 6 + 5 + 3 = 43
\]

### (f) \( n(A^c \cap B \cap C) \)
This refers to the elements that are outside of \( A \) but within both \( B \) and \( C \). Looking at the Venn diagram, the only region that fits this is the intersection of \( B \cap C \) that does not include \( A \), which is:

- \( B \cap C \): 5

So:
\[
n(A^c \cap B \cap C) = 5
\]

### Final answers:
- (a) \( n(A^c) = 22 \) (given)
- (b) \( n(A \cup B) = 23 \)
- (c) \( n(A \cap C^c) = 13 \)
- (d) \( n(A \cap B \cap C) = 3 \)
- (e) \( n(A \cup B \cup C) = 43 \)
- (f) \( n(A^c \cap B \cap C) = 5 \)

Do you want more details on any of these? 

Here are 5 related questions to consider:
1. How would the cardinalities change if an extra element were added to set \( A \cap B \cap C \)?
2. Can you compute \( n(B^c) \) from the Venn diagram?
3. What is \( n(C^c) \), and how does it relate to the other complements?
4. How would you calculate \( n(A \cap B^c \cap C) \)?
5. What is the significance of union and intersection in probability theory?

**Tip**: When analyzing Venn diagrams, carefully identify and sum the distinct regions for each operation (union, intersection, complement). This avoids double-counting elements.

Consider the Venn diagram shown below. Determine the following cardinalities: (a) n(A^c) = 22, (b) n(A ∪ B) = ?, (c) n(A ∩ C^c) = ?, (d) n(A ∩ B ∩ C) = ?, (e) n(A ∪ B ∪ C) = ?, (f) n(A^c ∩ B ∩ C) = ?

Explore a Venn diagram involving three sets (A, B, C) and calculate various cardinalities including union, intersection, and complement. This exercise covers key set theory concepts such as n(A ∪ B ∪ C), n(A ∩ B ∩ C), and more, suitable for high school students.

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