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Answer :
To solve this problem, we need to find the number of elements in the set [tex]\( A^C \cup B \)[/tex], where:
- [tex]\( n(U) = 200 \)[/tex]: Total number of elements in the universal set [tex]\( U \)[/tex].
- [tex]\( n(A) = 105 \)[/tex]: Number of elements in set [tex]\( A \)[/tex].
- [tex]\( n(B) = 115 \)[/tex]: Number of elements in set [tex]\( B \)[/tex].
- [tex]\( n(A \cap B) = 30 \)[/tex]: Number of elements in both sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
We use the principle of inclusion-exclusion to find [tex]\( n(A \cup B) \)[/tex], which is the number of elements in either set [tex]\( A \)[/tex] or set [tex]\( B \)[/tex], or in both:
[tex]\[
n(A \cup B) = n(A) + n(B) - n(A \cap B) = 105 + 115 - 30 = 190
\][/tex]
Next, we find the number of elements in the complement of [tex]\( A \)[/tex], denoted by [tex]\( A^C \)[/tex]. This is the set of elements that are in [tex]\( U \)[/tex] but not in [tex]\( A \)[/tex]:
[tex]\[
n(A^C) = n(U) - n(A) = 200 - 105 = 95
\][/tex]
Finally, we calculate [tex]\( n(A^C \cup B) \)[/tex], the number of elements in either [tex]\( A^C \)[/tex] or [tex]\( B \)[/tex], or in both. By applying the principle of inclusion-exclusion again in a similar way for complements, we get:
[tex]\[
n(A^C \cup B) = n(U)
\][/tex]
Thus, [tex]\( n(A^C \cup B) = 200 \)[/tex].
After solving, we find that the correct answer matches none of the given multiple choice options. Therefore, the answer is None of the above.
- [tex]\( n(U) = 200 \)[/tex]: Total number of elements in the universal set [tex]\( U \)[/tex].
- [tex]\( n(A) = 105 \)[/tex]: Number of elements in set [tex]\( A \)[/tex].
- [tex]\( n(B) = 115 \)[/tex]: Number of elements in set [tex]\( B \)[/tex].
- [tex]\( n(A \cap B) = 30 \)[/tex]: Number of elements in both sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
We use the principle of inclusion-exclusion to find [tex]\( n(A \cup B) \)[/tex], which is the number of elements in either set [tex]\( A \)[/tex] or set [tex]\( B \)[/tex], or in both:
[tex]\[
n(A \cup B) = n(A) + n(B) - n(A \cap B) = 105 + 115 - 30 = 190
\][/tex]
Next, we find the number of elements in the complement of [tex]\( A \)[/tex], denoted by [tex]\( A^C \)[/tex]. This is the set of elements that are in [tex]\( U \)[/tex] but not in [tex]\( A \)[/tex]:
[tex]\[
n(A^C) = n(U) - n(A) = 200 - 105 = 95
\][/tex]
Finally, we calculate [tex]\( n(A^C \cup B) \)[/tex], the number of elements in either [tex]\( A^C \)[/tex] or [tex]\( B \)[/tex], or in both. By applying the principle of inclusion-exclusion again in a similar way for complements, we get:
[tex]\[
n(A^C \cup B) = n(U)
\][/tex]
Thus, [tex]\( n(A^C \cup B) = 200 \)[/tex].
After solving, we find that the correct answer matches none of the given multiple choice options. Therefore, the answer is None of the above.
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