Use the given information to find the number of elements in each of the regions labeled R1 through R5.
n(A) = 29, n(B) = 34, n(C) = 39, n(A ∩ B) = 14, n(U) = 82

n(R1) = 	

 

n(R2) = 	

 

n(R3) = 	

 

n(R4) = 	

 

n(R5) =

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.

Use the given information to find the number of elements in each of the regions labeled R1 through R5.
n(A) = 29, n(B) = 34, n(C) = 39, n(A ∩ B) = 14, n(U) = 82

n(R1) = 

n(R2) = 

n(R3) = 

n(R4) = 

n(R5) =

Learn how to find the number of elements in regions labeled R1 through R5 using given data from sets A, B, C, and their intersections. This problem covers Venn diagrams and the inclusion-exclusion principle, ideal for students in Grades 8-10.

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