To find \( n(X \cup Y) \), the number of elements in the union of the sets \( X \) and \( Y \), we will use the principle of inclusion-exclusion for sets. The formula for the union of two finite sets is given by:
[tex]\[ n(X \cup Y) = n(X) + n(Y) - n(X \cap Y) \][/tex]
Here, \( n(X) \) represents the number of elements in set \( X \), \( n(Y) \) represents the number of elements in set \( Y \), and \( n(X \cap Y) \) represents the number of elements that are common to both sets \( X \) and \( Y \).
Given the values:
- \( n(X) = 12 \)
- \( n(Y) = 19 \)
- \( n(X \cap Y) = 7 \)
We substitute these values into the formula:
[tex]\[ n(X \cup Y) = 12 + 19 - 7 \][/tex]
Perform the arithmetic operations:
[tex]\[ n(X \cup Y) = 31 - 7 \][/tex]
[tex]\[ n(X \cup Y) = 24 \][/tex]
Thus, the number of elements in the union of sets \( X \) and \( Y \) is:
[tex]\[ n(X \cup Y) = 24 \][/tex]
So, the correct answer is:
(B) 24
