To find the value of [tex]\( n(A \cup B) \)[/tex], we can use the principle of inclusion-exclusion for sets. This principle is summarized in the formula:
[tex]\[
n(A \cup B) = n(A) + n(B) - n(A \cap B)
\][/tex]
Let's identify the given values:
- [tex]\( n(U) = 13 \)[/tex]: This is the size of the universal set, but it doesn't affect the calculation for [tex]\( n(A \cup B) \)[/tex].
- [tex]\( n(A) = 25 \)[/tex]: This is the number of elements in set [tex]\( A \)[/tex].
- [tex]\( n(B) = 18 \)[/tex]: This is the number of elements in set [tex]\( B \)[/tex].
- [tex]\( n(A \cap B) = 7 \)[/tex]: This is the number of elements that are in both sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
Now, substitute these values into the inclusion-exclusion formula:
[tex]\[
n(A \cup B) = 25 + 18 - 7
\][/tex]
Performing the arithmetic gives:
[tex]\[
n(A \cup B) = 43 - 7 = 36
\][/tex]
Thus, the value of [tex]\( n(A \cup B) \)[/tex] is [tex]\( 36 \)[/tex].
So, the correct answer is:
a) [tex]\( 36 \)[/tex]
