We are tasked with determining the number of elements in each region of the Venn diagram. The cardinalities provided are:

• $n(U) = 44$
• $n(A) = 18$
• $n(B) = 22$
• $n(C) = 18$
• $n(A \cap B) = 11$
• $n(A \cap C) = 8$
• $n(B \cap C) = 7$
• $n(A \cap B \cap C) = 3$ (already labeled in region V)

Step-by-Step Process:

We will calculate the number of elements in each region, using the Venn diagram labels:

Region V (already given as 3):

• Region V represents $A \cap B \cap C$, which is already provided as 3.

Region II (inside $A \cap B$ but not $C$):

• $n(A \cap B) = 11$, which includes Region V. Thus, Region II = $11 - 3 = 8$.

Region IV (inside $A \cap C$ but not $B$):

• $n(A \cap C) = 8$, which also includes Region V. Thus, Region IV = $8 - 3 = 5$.

Region VI (inside $B \cap C$ but not $A$):

• $n(B \cap C) = 7$, which also includes Region V. Thus, Region VI = $7 - 3 = 4$.

Region I (inside $A$ but not $B$ or $C$):

• $n(A) = 18$, which includes Regions II, IV, and V. Thus, Region I = $18 - (8 + 5 + 3) = 2$.

Region III (inside $B$ but not $A$ or $C$):

• $n(B) = 22$, which includes Regions II, VI, and V. Thus, Region III = $22 - (8 + 4 + 3) = 7$.

Region VII (inside $C$ but not $A$ or $B$):

• $n(C) = 18$, which includes Regions IV, VI, and V. Thus, Region VII = $18 - (5 + 4 + 3) = 6$.

Region VIII (outside all sets):

• $n(U) = 44$ represents the total number of elements. To find Region VIII, we subtract the sum of all other regions:

  $44 - (2 + 8 + 7 + 5 + 3 + 4 + 6) = 44 - 35 = 9$.

Final Answers:

• Region I = 2
• Region II = 8
• Region III = 7
• Region IV = 5
• Region VI = 4
• Region VII = 6
• Region VIII = 9

Would you like a detailed breakdown of these regions or further explanation?

Here are 5 follow-up questions to expand your understanding:

1. How do we calculate the number of elements in the union of sets $A$, $B$, and $C$?
2. What happens if two sets are disjoint? How would that change the diagram?
3. How can we verify that our solution is correct using the principle of inclusion-exclusion?
4. What is the significance of Region VIII in the context of the universal set $U$?
5. Can you think of a real-world example where Venn diagrams might be used to represent overlapping groups?

Tip: Always start by calculating the intersection regions first in Venn diagram problems, as they help clarify the more straightforward regions.