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Answer :
Let's solve the problem step-by-step using the provided information:
We are given:
- [tex]\( n(U) = 200 \)[/tex]
- [tex]\( n(A) = 105 \)[/tex]
- [tex]\( n(B) = 115 \)[/tex]
- [tex]\( n(A \cap B) = 30 \)[/tex]
We need to find [tex]\( n(A^c \cup B) \)[/tex].
### Step 1: Find [tex]\( n(A \cup B) \)[/tex]
We can use the formula for the union of two sets:
[tex]\[
n(A \cup B) = n(A) + n(B) - n(A \cap B)
\][/tex]
Substitute the given values:
[tex]\[
n(A \cup B) = 105 + 115 - 30 = 190
\][/tex]
### Step 2: Find [tex]\( n(A^c) \)[/tex]
The complement of set [tex]\( A \)[/tex] is all elements in the universal set [tex]\( U \)[/tex] that are not in [tex]\( A \)[/tex]. So,
[tex]\[
n(A^c) = n(U) - n(A)
\][/tex]
Substitute the known values:
[tex]\[
n(A^c) = 200 - 105 = 95
\][/tex]
### Step 3: Find [tex]\( n(A^c \cup B) \)[/tex]
By definition, [tex]\( A^c \cup B \)[/tex] includes all elements that are either not in [tex]\( A \)[/tex] or are in [tex]\( B \)[/tex]. We find this by using the union of the complement of [tex]\( A \)[/tex] with [tex]\( B \)[/tex]:
From the previous steps, we have already found that:
- [tex]\( n(A^c) = 95 \)[/tex]
- [tex]\( n(A \cup B) = 190 \)[/tex]
Now, using these, the total number of elements in [tex]\( A^c \cup B \)[/tex] is equal to the total number of elements in the universal set, [tex]\( n(U) \)[/tex], since [tex]\( A^c \cup B \)[/tex] encompasses all elements in [tex]\( U \)[/tex].
[tex]\[
n(A^c \cup B) = 200
\][/tex]
Thus, the correct answer is [tex]\( 200 \)[/tex], which is not listed among the options. Therefore, the correct answer is None of the above.
We are given:
- [tex]\( n(U) = 200 \)[/tex]
- [tex]\( n(A) = 105 \)[/tex]
- [tex]\( n(B) = 115 \)[/tex]
- [tex]\( n(A \cap B) = 30 \)[/tex]
We need to find [tex]\( n(A^c \cup B) \)[/tex].
### Step 1: Find [tex]\( n(A \cup B) \)[/tex]
We can use the formula for the union of two sets:
[tex]\[
n(A \cup B) = n(A) + n(B) - n(A \cap B)
\][/tex]
Substitute the given values:
[tex]\[
n(A \cup B) = 105 + 115 - 30 = 190
\][/tex]
### Step 2: Find [tex]\( n(A^c) \)[/tex]
The complement of set [tex]\( A \)[/tex] is all elements in the universal set [tex]\( U \)[/tex] that are not in [tex]\( A \)[/tex]. So,
[tex]\[
n(A^c) = n(U) - n(A)
\][/tex]
Substitute the known values:
[tex]\[
n(A^c) = 200 - 105 = 95
\][/tex]
### Step 3: Find [tex]\( n(A^c \cup B) \)[/tex]
By definition, [tex]\( A^c \cup B \)[/tex] includes all elements that are either not in [tex]\( A \)[/tex] or are in [tex]\( B \)[/tex]. We find this by using the union of the complement of [tex]\( A \)[/tex] with [tex]\( B \)[/tex]:
From the previous steps, we have already found that:
- [tex]\( n(A^c) = 95 \)[/tex]
- [tex]\( n(A \cup B) = 190 \)[/tex]
Now, using these, the total number of elements in [tex]\( A^c \cup B \)[/tex] is equal to the total number of elements in the universal set, [tex]\( n(U) \)[/tex], since [tex]\( A^c \cup B \)[/tex] encompasses all elements in [tex]\( U \)[/tex].
[tex]\[
n(A^c \cup B) = 200
\][/tex]
Thus, the correct answer is [tex]\( 200 \)[/tex], which is not listed among the options. Therefore, the correct answer is None of the above.
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