To find the least value of [tex]\( n(A \cup B) \)[/tex], where [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are two sets, we can use the principle of set theory related to the union of two sets.
Given:
- [tex]\( n(A) = 80 \)[/tex]
- [tex]\( M(B) = 65 \)[/tex]
The least value of [tex]\( n(A \cup B) \)[/tex] occurs when there is no intersection between sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex]. In other words, when [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are disjoint sets.
For disjoint sets, the number of elements in the union is simply the sum of the number of elements in each set. Therefore, we perform the following:
[tex]\[ n(A \cup B) = n(A) + M(B) \][/tex]
Substituting the given values:
[tex]\[ n(A \cup B) = 80 + 65 \][/tex]
[tex]\[ n(A \cup B) = 145 \][/tex]
Thus, the least value of [tex]\( n(A \cup B) \)[/tex] is [tex]\( 145 \)[/tex].
None of the given options—125, 20, 80, or 65—are correct based on this calculation. The correct answer is [tex]\( 145 \)[/tex].
