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.