Get reliable answers to your questions at Westonci.ca, where our knowledgeable community is always ready to help. Join our Q&A platform and connect with professionals ready to provide precise answers to your questions in various areas. 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.
We appreciate your time on our site. Don't hesitate to return whenever you have more questions or need further clarification. Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.
