To find the total number of elements in the universe (denoted as [tex]\( n(U) \)[/tex]), we are given the following information:
- [tex]\( n(A') = 25 \)[/tex]
- [tex]\( n(B) = 24 \)[/tex]
- [tex]\( n(A' \cup B') = 37 \)[/tex]
- [tex]\( n(A \cap B) = 10 \)[/tex]
First, let's recall a few set theory principles and relationships:
1. [tex]\( n(A') \)[/tex] is the number of elements not in set [tex]\( A \)[/tex].
2. [tex]\( n(A \cup B) \)[/tex] is the number of elements in either set [tex]\( A \)[/tex] or [tex]\( B \)[/tex] or both.
3. [tex]\( n(A' \cup B') \)[/tex] is the number of elements in either [tex]\( A' \)[/tex] (not in [tex]\( A \)[/tex]) or [tex]\( B' \)[/tex] (not in [tex]\( B \)[/tex]) or both. It can be related to the elements that are in neither [tex]\( A \)[/tex] nor [tex]\( B \)[/tex].
To solve for [tex]\( n(U) \)[/tex], we proceed as follows:
1. Using the De Morgan's law, we know:
[tex]\[
n(A' \cup B') = n(U) - n(A \cap B)
\][/tex]
Given [tex]\( n(A' \cup B') = 37 \)[/tex]:
[tex]\[
37 = n(U) - n(A \cap B)
\][/tex]
Hence:
[tex]\[
37 = n(U) - 10
\][/tex]
Solving for [tex]\( n(U) \)[/tex]:
[tex]\[
n(U) = 37 + 10
\][/tex]
[tex]\[
n(U) = 47
\][/tex]
Therefore, the total number of elements in the universe is [tex]\( 47 \)[/tex].
