Find the best solutions to your questions at Westonci.ca, the premier Q&A platform with a community of knowledgeable experts. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.
Sagot :
Sure, let's go through this step-by-step.
### a. Define the set
Let [tex]\( A \)[/tex] be the set with [tex]\( n(A) = 40 \)[/tex] elements.
Let [tex]\( B \)[/tex] be the set with [tex]\( n(B) = 60 \)[/tex] elements.
* The union of sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex], denoted by [tex]\( A \cup B \)[/tex], contains [tex]\( n(A \cup B) = 80 \)[/tex] elements.
### b. Find the value of [tex]\( n(A \cap B) \)[/tex]
To find the number of elements in the intersection of sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex], denoted by [tex]\( n(A \cap B) \)[/tex], we use the formula for the union of two sets:
[tex]\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \][/tex]
Plugging the given values into the formula, we get:
[tex]\[ 80 = 40 + 60 - n(A \cap B) \][/tex]
Simplifying the equation:
[tex]\[ 80 = 100 - n(A \cap B) \][/tex]
Solving for [tex]\( n(A \cap B) \)[/tex]:
[tex]\[ n(A \cap B) = 100 - 80 \][/tex]
[tex]\[ n(A \cap B) = 20 \][/tex]
Thus, [tex]\( n(A \cap B) = 20 \)[/tex].
### c. Find the value of [tex]\( n(A) - n(A \cap B) \)[/tex]
This is the number of elements only in set [tex]\( A \)[/tex], i.e., the elements in [tex]\( A \)[/tex] that are not in [tex]\( B \)[/tex]:
[tex]\[ n(\text{only } A) = n(A) - n(A \cap B) \][/tex]
Given [tex]\( n(A) = 40 \)[/tex] and [tex]\( n(A \cap B) = 20 \)[/tex]:
[tex]\[ n(\text{only } A) = 40 - 20 \][/tex]
[tex]\[ n(\text{only } A) = 20 \][/tex]
Thus, the number of elements only in [tex]\( A \)[/tex] is [tex]\( 20 \)[/tex].
### d. Represent the above information in a Venn diagram
To represent the given information in a Venn diagram:
1. Draw two intersecting circles, one representing set [tex]\( A \)[/tex] and the other representing set [tex]\( B \)[/tex].
2. The intersection (common area) of these circles represents [tex]\( A \cap B \)[/tex] and contains [tex]\( 20 \)[/tex] elements.
3. The part of circle [tex]\( A \)[/tex] excluding the intersection represents the elements only in [tex]\( A \)[/tex] and contains [tex]\( 20 \)[/tex] elements.
4. The part of circle [tex]\( B \)[/tex] excluding the intersection is for the elements only in [tex]\( B \)[/tex]. Given [tex]\( n(B) = 60 \)[/tex] and [tex]\( n(A \cap B) = 20 \)[/tex], this part contains [tex]\( 60 - 20 = 40 \)[/tex] elements.
5. The total number of elements in [tex]\( A \cup B \)[/tex] (the union of sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex]) is consistent with the given [tex]\( 80 \)[/tex] elements.
Graphically, the Venn diagram would look something like this:
```
_______________
/ \
/ 20 \
| (__20__) 40 |
| A |
|\______________ /
\_____________ / B
```
In this Venn diagram:
- The left circle (A) contains 20 elements that are only in A.
- The intersection of both circles contains the 20 elements common to A and B.
- The right circle (B) outside the intersection area contains 40 elements that are only in B.
### a. Define the set
Let [tex]\( A \)[/tex] be the set with [tex]\( n(A) = 40 \)[/tex] elements.
Let [tex]\( B \)[/tex] be the set with [tex]\( n(B) = 60 \)[/tex] elements.
* The union of sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex], denoted by [tex]\( A \cup B \)[/tex], contains [tex]\( n(A \cup B) = 80 \)[/tex] elements.
### b. Find the value of [tex]\( n(A \cap B) \)[/tex]
To find the number of elements in the intersection of sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex], denoted by [tex]\( n(A \cap B) \)[/tex], we use the formula for the union of two sets:
[tex]\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \][/tex]
Plugging the given values into the formula, we get:
[tex]\[ 80 = 40 + 60 - n(A \cap B) \][/tex]
Simplifying the equation:
[tex]\[ 80 = 100 - n(A \cap B) \][/tex]
Solving for [tex]\( n(A \cap B) \)[/tex]:
[tex]\[ n(A \cap B) = 100 - 80 \][/tex]
[tex]\[ n(A \cap B) = 20 \][/tex]
Thus, [tex]\( n(A \cap B) = 20 \)[/tex].
### c. Find the value of [tex]\( n(A) - n(A \cap B) \)[/tex]
This is the number of elements only in set [tex]\( A \)[/tex], i.e., the elements in [tex]\( A \)[/tex] that are not in [tex]\( B \)[/tex]:
[tex]\[ n(\text{only } A) = n(A) - n(A \cap B) \][/tex]
Given [tex]\( n(A) = 40 \)[/tex] and [tex]\( n(A \cap B) = 20 \)[/tex]:
[tex]\[ n(\text{only } A) = 40 - 20 \][/tex]
[tex]\[ n(\text{only } A) = 20 \][/tex]
Thus, the number of elements only in [tex]\( A \)[/tex] is [tex]\( 20 \)[/tex].
### d. Represent the above information in a Venn diagram
To represent the given information in a Venn diagram:
1. Draw two intersecting circles, one representing set [tex]\( A \)[/tex] and the other representing set [tex]\( B \)[/tex].
2. The intersection (common area) of these circles represents [tex]\( A \cap B \)[/tex] and contains [tex]\( 20 \)[/tex] elements.
3. The part of circle [tex]\( A \)[/tex] excluding the intersection represents the elements only in [tex]\( A \)[/tex] and contains [tex]\( 20 \)[/tex] elements.
4. The part of circle [tex]\( B \)[/tex] excluding the intersection is for the elements only in [tex]\( B \)[/tex]. Given [tex]\( n(B) = 60 \)[/tex] and [tex]\( n(A \cap B) = 20 \)[/tex], this part contains [tex]\( 60 - 20 = 40 \)[/tex] elements.
5. The total number of elements in [tex]\( A \cup B \)[/tex] (the union of sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex]) is consistent with the given [tex]\( 80 \)[/tex] elements.
Graphically, the Venn diagram would look something like this:
```
_______________
/ \
/ 20 \
| (__20__) 40 |
| A |
|\______________ /
\_____________ / B
```
In this Venn diagram:
- The left circle (A) contains 20 elements that are only in A.
- The intersection of both circles contains the 20 elements common to A and B.
- The right circle (B) outside the intersection area contains 40 elements that are only in B.
Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Westonci.ca is your trusted source for answers. Visit us again to find more information on diverse topics.
