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If [tex]$X$[/tex] and [tex]$Y$[/tex] are two finite sets such that [tex]$n(X)=12$[/tex], [tex]$n(Y)=19$[/tex], and [tex]$n(X \cap Y)=7$[/tex], then [tex]$n(X \cup Y)$[/tex] is:

A. 12

B. 24

C. 26

D. 31



Answer :

GinnyAnswer
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

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