Let's work through the Venn diagram and answer the questions. The sets $A$, $B$, and $C$ are represented by circles with different numbers inside each region, corresponding to specific intersections and unions of the sets.

The values in the Venn diagram:

• Region $A$ only: 4
• Region $B$ only: 5
• Region $C$ only: 7
• $A \cap B$ (excluding $C$): 6
• $A \cap C$ (excluding $B$): 1
• $B \cap C$ (excluding $A$): 10
• $A \cap B \cap C$: 8
• Outside $A \cup B \cup C$: 12

Now, let's determine the following cardinalities:

1. $n(\bar{A})$ (the number of elements outside $A$):

   • Everything outside $A$ includes the values from regions not part of $A$, which are: $5 \, (B \text{ only}) + 10 \, (B \cap C) + 7 \, (C \text{ only}) + 12 \, (\text{outside all sets}) = 34$ Therefore, $n(\bar{A}) = 34$.

2. $n(A \cup B)$ (the union of sets $A$ and $B$):

   • Add all the regions that are part of $A$, $B$, or both: $4 \, (A \text{ only}) + 6 \, (A \cap B) + 8 \, (A \cap B \cap C) + 5 \, (B \text{ only}) + 1 \, (A \cap C) = 24$ Therefore, $n(A \cup B) = 24$.

3. $n(A \cap C)$ (the intersection of sets $A$ and $C$):

   • Add the regions that are part of both $A$ and $C$: $1 \, (A \cap C \text{ only}) + 8 \, (A \cap B \cap C) = 9$ Therefore, $n(A \cap C) = 9$.

4. $n(A \cap B \cap C)$ (the intersection of $A$, $B$, and $C$):

   • This region is given as $8$. Therefore, $n(A \cap B \cap C) = 8$.

5. $n(\bar{A} \cup B \cup C)$ (the union of everything outside $A$, and the sets $B$ and $C$):

   • This would be everything in $B$, $C$, or outside $A$: $5 \, (B \text{ only}) + 10 \, (B \cap C) + 7 \, (C \text{ only}) + 12 \, (\text{outside all sets}) = 34$ Therefore, $n(\bar{A} \cup B \cup C) = 34$.

Summary:

1. $n(\bar{A}) = 34$
2. $n(A \cup B) = 24$
3. $n(A \cap C) = 9$
4. $n(A \cap B \cap C) = 8$
5. $n(\bar{A} \cup B \cup C) = 34$

Would you like further details or explanations? Here are some follow-up questions to consider:

1. How would the calculation change if an additional region was added to the Venn diagram?
2. Can we find $n(A \cap B)$ from this diagram?
3. How would $n(\bar{B})$ be calculated from the Venn diagram?
4. Can you determine the total number of elements in the universal set?
5. What is the formula for the union of three sets, and how would it apply here?

Tip: When calculating the union of two sets, remember to subtract any intersections that were counted twice.