The accompanying Venn diagram shows the number of elements in region V. Use the given cardinalities to determine the number of elements in each of the other seven regions.

n(U) = 44, n(A) = 18, n(B) = 23,

n(C) = 26, n(AnB) = 9,

n(ANC) = 11, n(BC) = 13

U

IV

VII

There are

elements in region I.

There are

elements in region II.

There are

elements in region Ill.

There are

elements in region IV.

There are

elements in region VI.

There are

elements in region VIl.

There are

elements in region VIII.


To solve this problem, we need to determine the number of elements in each region of the Venn diagram. The regions correspond to the different parts of the sets \( A \), \( B \), and \( C \), as well as the universal set \( U \). We'll use the given cardinalities of the sets and their intersections to determine the number of elements in each of the eight regions.

### Given Information:
- \( n(U) = 44 \) (total number of elements in the universal set)
- \( n(A) = 18 \) (number of elements in set \( A \))
- \( n(B) = 23 \) (number of elements in set \( B \))
- \( n(C) = 26 \) (number of elements in set \( C \))
- \( n(A \cap B) = 9 \) (number of elements in the intersection of \( A \) and \( B \))
- \( n(A \cap C) = 11 \) (number of elements in the intersection of \( A \) and \( C \))
- \( n(B \cap C) = 13 \) (number of elements in the intersection of \( B \) and \( C \))

Let's denote each region of the Venn diagram as follows:
1. Region I: Elements in \( A \) only.
2. Region II: Elements in \( A \cap B \), but not in \( C \).
3. Region III: Elements in \( A \cap C \), but not in \( B \).
4. Region IV: Elements in \( A \cap B \cap C \) (all three sets).
5. Region V: Elements in \( B \) only.
6. Region VI: Elements in \( B \cap C \), but not in \( A \).
7. Region VII: Elements in \( C \) only.
8. Region VIII: Elements outside all three sets, but in \( U \).

### Step-by-Step Solution:

1. **Region IV (\( A \cap B \cap C \)):**
   - This is the number of elements common to all three sets, and it is included in all the intersections.
   - Let's denote this as \( x \).
   - We will calculate it last after checking the individual and two-set intersections.

2. **Region II (\( A \cap B \), but not in \( C \)):**
   - The total number of elements in \( A \cap B \) is 9.
   - Out of these, some elements are also in \( C \). So, subtract the elements in \( A \cap B \cap C \) (denoted as \( x \)).
   - Region II = \( 9 - x \).

3. **Region III (\( A \cap C \), but not in \( B \)):**
   - The total number of elements in \( A \cap C \) is 11.
   - Subtract the elements in \( A \cap B \cap C \) (denoted as \( x \)).
   - Region III = \( 11 - x \).

4. **Region VI (\( B \cap C \), but not in \( A \)):**
   - The total number of elements in \( B \cap C \) is 13.
   - Subtract the elements in \( A \cap B \cap C \) (denoted as \( x \)).
   - Region VI = \( 13 - x \).

5. **Region I (\( A \) only):**
   - The total number of elements in \( A \) is 18.
   - Subtract the elements that are also in \( B \), \( C \), and \( A \cap B \cap C \).
   - Region I = \( 18 - ((9 - x) + (11 - x) + x) \).

6. **Region V (\( B \) only):**
   - The total number of elements in \( B \) is 23.
   - Subtract the elements that are also in \( A \), \( C \), and \( A \cap B \cap C \).
   - Region V = \( 23 - ((9 - x) + (13 - x) + x) \).

7. **Region VII (\( C \) only):**
   - The total number of elements in \( C \) is 26.
   - Subtract the elements that are also in \( A \), \( B \), and \( A \cap B \cap C \).
   - Region VII = \( 26 - ((11 - x) + (13 - x) + x) \).

8. **Region VIII (Outside all three sets):**
   - The total number of elements in \( U \) is 44.
   - Subtract the elements in \( A \), \( B \), and \( C \) (all regions of the Venn diagram).
   - Region VIII = \( 44 - (Region I + Region II + Region III + Region IV + Region V + Region VI + Region VII) \).

Once you solve for \( x \), substitute it back into the equations for the other regions. Would you like me to proceed with calculating \( x \) and the other regions?

The accompanying Venn diagram shows the number of elements in region V. Use the given cardinalities to determine the number of elements in each of the other seven regions.

n(U) = 44, n(A) = 18, n(B) = 23, n(C) = 26, n(AnB) = 9, n(ANC) = 11, n(BC) = 13

U

IV

VII

There are elements in region I.

There are elements in region II.

There are elements in region Ill.

There are elements in region IV.

There are elements in region VI.

There are elements in region VIl.

There are elements in region VIII.

Solve a Venn diagram problem by determining the number of elements in each region using set theory and cardinalities. The problem involves 3 sets A, B, and C, and uses the inclusion-exclusion principle. Given n(U) = 44, n(A) = 18, n(B) = 23, n(C) = 26, find the elements in each region of the diagram.

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