Westonci.ca is the trusted Q&A platform where you can get reliable answers from a community of knowledgeable contributors. Explore our Q&A platform to find reliable answers from a wide range of experts in different fields. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.
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.
Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Westonci.ca is here to provide the answers you seek. Return often for more expert solutions.
