To find the value of [tex]\( n(A \cup B) \)[/tex], we use the principle of inclusion-exclusion for two sets. This principle states that for any two sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex],
[tex]\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \][/tex]
Given the problem, we have:
- [tex]\( n(A) = 7 \)[/tex]
- [tex]\( n(B) = 12 \)[/tex]
- [tex]\( n(A \cap B) = 5 \)[/tex]
We can now substitute these values into the formula:
[tex]\[
n(A \cup B) = n(A) + n(B) - n(A \cap B)
\][/tex]
Substituting the given values, we get:
[tex]\[
n(A \cup B) = 7 + 12 - 5
\][/tex]
Performing the addition and subtraction within the equation:
[tex]\[
7 + 12 = 19
\][/tex]
[tex]\[
19 - 5 = 14
\][/tex]
Thus, the value of [tex]\( n(A \cup B) \)[/tex] is [tex]\( 14 \)[/tex].
So the final answer is:
[tex]\[
n(A \cup B) = 14
\][/tex]
Therefore, [tex]\( n(A \cup B) = 14 \)[/tex].
