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Find the value of [tex]\( n(A \cup B) \)[/tex] if [tex]\( n(A) = 7 \)[/tex], [tex]\( n(B) = 12 \)[/tex], and [tex]\( n(A \cap B) = 5 \)[/tex].

[tex]\[ n(A \cup B) = \][/tex]
[tex]\[ \square \][/tex]
(Type a whole number.)

Sagot :

GinnyAnswer
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].
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