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Sagot :
Alright, let's approach this problem methodically. Given the information:
- [tex]\( n(U) = 210 \)[/tex]
- [tex]\( n(A) = 120 \)[/tex]
- [tex]\( n(B) = 60 \)[/tex]
- [tex]\( n((A \cup B)') = 45 \)[/tex]
We need to find:
- The Venn diagram representation.
- The value of only [tex]\( A \)[/tex].
- The value of [tex]\( A \cap B \)[/tex].
### Step-by-Step Solution:
#### (a) Venn Diagram Representation
1. Universal Set [tex]\( U \)[/tex]: This describes the entire set, which contains all elements we're considering. The total number of elements in [tex]\( U \)[/tex] is 210.
2. Subset [tex]\( A \)[/tex]: A subset of [tex]\( U \)[/tex] with 120 elements ( [tex]\( n(A) = 120 \)[/tex] ).
3. Subset [tex]\( B \)[/tex]: Another subset of [tex]\( U \)[/tex] with 60 elements ( [tex]\( n(B) = 60 \)[/tex] ).
4. Complement of [tex]\( A \cup B \)[/tex]: The number of elements not in [tex]\( A \cup B \)[/tex] is 45 ( [tex]\( n((A \cup B)') = 45 \)[/tex] ).
The point is to show the relationships and overlapping sections in the Venn diagram:
- Let the part of [tex]\( A \)[/tex] that does not intersect [tex]\( B \)[/tex] be [tex]\( A_{\text{only}} \)[/tex].
- Let the part of [tex]\( B \)[/tex] that does not intersect [tex]\( A \)[/tex] be [tex]\( B_{\text{only}} \)[/tex].
- Let the overlapping part (intersection of [tex]\( A \)[/tex] and [tex]\( B \)[/tex]) be represented as [tex]\( A \cap B \)[/tex].
#### (b) Finding the Value of Only [tex]\( A \)[/tex]
To find only [tex]\( A \)[/tex], we need to determine the value of [tex]\( A \)[/tex] excluding the part that intersects with [tex]\( B \)[/tex].
The values calculated:
1. Calculate [tex]\( n(A \cup B) \)[/tex]:
- Since [tex]\( n((A \cup B)') = 45 \)[/tex], the number of elements in [tex]\( A \cup B \)[/tex] can be found from:
[tex]\[ n(A \cup B) = n(U) - n((A \cup B)') \][/tex]
[tex]\[ n(A \cup B) = 210 - 45 \][/tex]
[tex]\[ n(A \cup B) = 165 \][/tex]
2. Calculate [tex]\( n(A \cap B) \)[/tex] using the principle of inclusion-exclusion:
[tex]\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \][/tex]
Solving for [tex]\( n(A \cap B) \)[/tex]:
[tex]\[ 165 = 120 + 60 - n(A \cap B) \][/tex]
[tex]\[ 165 = 180 - n(A \cap B) \][/tex]
[tex]\[ n(A \cap B) = 180 - 165 \][/tex]
[tex]\[ n(A \cap B) = 15 \][/tex]
3. Calculate the Value of Only [tex]\( A \)[/tex]:
[tex]\[ \text{Only } A = n(A) - n(A \cap B) \][/tex]
[tex]\[ \text{Only } A = 120 - 15 \][/tex]
[tex]\[ \text{Only } A = 105 \][/tex]
#### (c) Finding the Value of [tex]\( A \cap B \)[/tex]
From our calculations above, we found that:
[tex]\[ n(A \cap B) = 15 \][/tex]
### Conclusion:
- (a): The Venn diagram would depict the total universal set [tex]\( U \)[/tex], with two overlapping subsets [tex]\( A \)[/tex] and [tex]\( B \)[/tex]. [tex]\( A \cup B \)[/tex]'s complement contains 45 elements, meaning 165 elements fall within either [tex]\( A \)[/tex] or [tex]\( B \)[/tex] or their intersection.
- (b): The value of only [tex]\( A \)[/tex] is [tex]\( \boxed{105} \)[/tex].
- (c): The value of [tex]\( A \cap B \)[/tex] is [tex]\( \boxed{15} \)[/tex].
Thus, the calculations are neatly confirmed.
- [tex]\( n(U) = 210 \)[/tex]
- [tex]\( n(A) = 120 \)[/tex]
- [tex]\( n(B) = 60 \)[/tex]
- [tex]\( n((A \cup B)') = 45 \)[/tex]
We need to find:
- The Venn diagram representation.
- The value of only [tex]\( A \)[/tex].
- The value of [tex]\( A \cap B \)[/tex].
### Step-by-Step Solution:
#### (a) Venn Diagram Representation
1. Universal Set [tex]\( U \)[/tex]: This describes the entire set, which contains all elements we're considering. The total number of elements in [tex]\( U \)[/tex] is 210.
2. Subset [tex]\( A \)[/tex]: A subset of [tex]\( U \)[/tex] with 120 elements ( [tex]\( n(A) = 120 \)[/tex] ).
3. Subset [tex]\( B \)[/tex]: Another subset of [tex]\( U \)[/tex] with 60 elements ( [tex]\( n(B) = 60 \)[/tex] ).
4. Complement of [tex]\( A \cup B \)[/tex]: The number of elements not in [tex]\( A \cup B \)[/tex] is 45 ( [tex]\( n((A \cup B)') = 45 \)[/tex] ).
The point is to show the relationships and overlapping sections in the Venn diagram:
- Let the part of [tex]\( A \)[/tex] that does not intersect [tex]\( B \)[/tex] be [tex]\( A_{\text{only}} \)[/tex].
- Let the part of [tex]\( B \)[/tex] that does not intersect [tex]\( A \)[/tex] be [tex]\( B_{\text{only}} \)[/tex].
- Let the overlapping part (intersection of [tex]\( A \)[/tex] and [tex]\( B \)[/tex]) be represented as [tex]\( A \cap B \)[/tex].
#### (b) Finding the Value of Only [tex]\( A \)[/tex]
To find only [tex]\( A \)[/tex], we need to determine the value of [tex]\( A \)[/tex] excluding the part that intersects with [tex]\( B \)[/tex].
The values calculated:
1. Calculate [tex]\( n(A \cup B) \)[/tex]:
- Since [tex]\( n((A \cup B)') = 45 \)[/tex], the number of elements in [tex]\( A \cup B \)[/tex] can be found from:
[tex]\[ n(A \cup B) = n(U) - n((A \cup B)') \][/tex]
[tex]\[ n(A \cup B) = 210 - 45 \][/tex]
[tex]\[ n(A \cup B) = 165 \][/tex]
2. Calculate [tex]\( n(A \cap B) \)[/tex] using the principle of inclusion-exclusion:
[tex]\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \][/tex]
Solving for [tex]\( n(A \cap B) \)[/tex]:
[tex]\[ 165 = 120 + 60 - n(A \cap B) \][/tex]
[tex]\[ 165 = 180 - n(A \cap B) \][/tex]
[tex]\[ n(A \cap B) = 180 - 165 \][/tex]
[tex]\[ n(A \cap B) = 15 \][/tex]
3. Calculate the Value of Only [tex]\( A \)[/tex]:
[tex]\[ \text{Only } A = n(A) - n(A \cap B) \][/tex]
[tex]\[ \text{Only } A = 120 - 15 \][/tex]
[tex]\[ \text{Only } A = 105 \][/tex]
#### (c) Finding the Value of [tex]\( A \cap B \)[/tex]
From our calculations above, we found that:
[tex]\[ n(A \cap B) = 15 \][/tex]
### Conclusion:
- (a): The Venn diagram would depict the total universal set [tex]\( U \)[/tex], with two overlapping subsets [tex]\( A \)[/tex] and [tex]\( B \)[/tex]. [tex]\( A \cup B \)[/tex]'s complement contains 45 elements, meaning 165 elements fall within either [tex]\( A \)[/tex] or [tex]\( B \)[/tex] or their intersection.
- (b): The value of only [tex]\( A \)[/tex] is [tex]\( \boxed{105} \)[/tex].
- (c): The value of [tex]\( A \cap B \)[/tex] is [tex]\( \boxed{15} \)[/tex].
Thus, the calculations are neatly confirmed.
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