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.