To solve this problem, we need to determine the number of elements in each region $R_1$ through $R_5$, using the given information about sets $A$, $B$, and $C$ in a universal set $U$. These sets are likely overlapping, and we can visualize this using a Venn diagram.

Given dataTo solve this problem, we need to determine the number of elements in each region $R_1$ through $R_5$, using the given information about sets $A$, $B$, and $C$ in a universal set $U$. These sets are likely overlapping, and we can visualize this using a Venn diagram.

Given

• $n(A) = 29$ (Number of elements in set $A$)
• $n(B) = 34$ (Number of elements in set $B$)
• $n(C) = 39$ (Number of elements in set $C$)
• $n(A \cap B) = 14$ (Number of elements common to both $A$ and $B$)
• $n(U) = 82$ (Total number of elements in the universal set)

We also define the regions in a typical three-set Venn diagram as follows:

• $R_1$: Elements in set $A$ only (not in $B$ or $C$)
• $R_2$: Elements in set $B$ only (not in $A$ or $C$)
• $R_3$: Elements in set $C$ only (not in $A$ or $B$)
• $R_4$: Elements in both $A$ and $B$ (but not in $C$)
• $R_5$: Elements in neither $A$, $B$, nor $C$ (complement of $A \cup B \cup C$)

Step-by-step process:

1. Finding $R_4$:

   • $R_4$ is the intersection of $A$ and $B$, but not $C$. We are given $n(A \cap B) = 14$, so $R_4 = 14$.

2. Finding $R_1$ and $R_2$:

   • From $n(A) = 29$ and $n(B) = 34$, and knowing that $R_4 = 14$ (common elements), we subtract these from the total elements of $A$ and $B$.

   • $n(R_1) = n(A) - n(A \cap B) = 29 - 14 = 15$.

   • $n(R_2) = n(B) - n(A \cap B) = 34 - 14 = 20$.

3. Finding $R_3$:

   • $R_3$ is the part of $C$ only, and since we don't have specific information about the intersection with $A$ or $B$, we assume $R_3$ remains at $n(C) = 39$.

4. Finding $R_5$:

   • $R_5$ is the part of the universal set that is not in $A$, $B$, or $C$.
   • Total elements in $A \cup B \cup C$ = $n(A) + n(B) + n(C) - n(A \cap B)$.
     • $= 29 + 34 + 39 - 14 = 88$.
   • Since $n(U) = 82$, the number of elements in $R_5 = n(U) - n(A \cup B \cup C) = 82 - 88$.

Let me know if you have questions.