Questions: 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) = 25, n(~B) = 26 n(C) = 18, n(~A ∩ ~B) = 13 n(~A ∩ C) = 8, n(~B ∩ C) = 11 There are elements in region I. There are elements in region III. There are elements in region IV. There are elements in region VI. There are elements in region VII. elements in region VIII.

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) = 25, n(~B) = 26
n(C) = 18, n(~A ∩ ~B) = 13
n(~A ∩ C) = 8, n(~B ∩ C) = 11

There are elements in region I.
There are elements in region III.
There are elements in region IV.
There are elements in region VI.
There are elements in region VII. elements in region VIII.

Transcript text: 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. \[ \begin{array}{l} n(\mathrm{U})=44, n(\mathrm{~A})=25, n(\mathrm{~B})=26 \\ n(\mathrm{C})=18, n(\mathrm{~A} \cap \mathrm{B})=13 \\ n(\mathrm{~A} \cap \mathrm{C})=8, n(\mathrm{~B} \cap \mathrm{C})=11 \end{array} \] There are $\square$ elements in region I. There are $\square$ elements in region III. There are $\square$ elements in region IV. There are $\square$ elements in region VI. There are $\square$ elements in region VII. elements in region VIII.

Solution

Solution Steps

Step 1: Identify the given values

The problem provides the following values:

$ n(U) = 44 $
$ n(A) = 25 $
$ n(B) = 26 $
$ n(C) = 18 $
$ n(A \cap B) = 13 $
$ n(A \cap C) = 8 $
$ n(B \cap C) = 11 $
$ n(A \cap B \cap C) = 5 $

Step 2: Use the principle of inclusion-exclusion

To find the number of elements in each region, we use the principle of inclusion-exclusion for three sets: \[ n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(A \cap C) - n(B \cap C) + n(A \cap B \cap C) \]

Step 3: Calculate $ n(A \cup B \cup C) $

Substitute the given values into the formula: \[ n(A \cup B \cup C) = 25 + 26 + 18 - 13 - 8 - 11 + 5 = 42 \]

Step 4: Determine the number of elements in region VIII

Region VIII represents the elements in $ U $ that are not in $ A \cup B \cup C $: \[ n(\text{Region VIII}) = n(U) - n(A \cup B \cup C) = 44 - 42 = 2 \]

Step 5: Determine the number of elements in regions I, II, and III

Using the given values and the Venn diagram:

Region I (only in $ A $): $ n(A) - n(A \cap B) - n(A \cap C) + n(A \cap B \cap C) = 25 - 13 - 8 + 5 = 9 $
Region II (only in $ B $): $ n(B) - n(A \cap B) - n(B \cap C) + n(A \cap B \cap C) = 26 - 13 - 11 + 5 = 7 $
Region III (only in $ C $): $ n(C) - n(A \cap C) - n(B \cap C) + n(A \cap B \cap C) = 18 - 8 - 11 + 5 = 4 $

Step 6: Determine the number of elements in regions IV, V, VI, and VII

Using the given values and the Venn diagram:

Region IV (in $ A \cap B $ but not $ C $): $ n(A \cap B) - n(A \cap B \cap C) = 13 - 5 = 8 $
Region V (in $ A \cap C $ but not $ B $): $ n(A \cap C) - n(A \cap B \cap C) = 8 - 5 = 3 $
Region VI (in $ B \cap C $ but not $ A $): $ n(B \cap C) - n(A \cap B \cap C) = 11 - 5 = 6 $
Region VII (in $ A \cap B \cap C $): $ n(A \cap B \cap C) = 5 $