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Given that [tex]$A$[/tex] and [tex]$B$[/tex] are two subsets of a universal set [tex]$U$[/tex] in which [tex]$n(U)=100$[/tex], [tex]$n(A)=60$[/tex], [tex]$n(B)=40$[/tex], and [tex]$n(\overline{A \cup B})=5$[/tex], find the value of [tex]$n(A \cap B)$[/tex].

Sagot :

GinnyAnswer
To find the value of [tex]\( n(A \cap B) \)[/tex], we need to use the principles of set theory as well as the given values.

1. Identify the Given Values:
- The size of the universal set [tex]\( U \)[/tex]: [tex]\( n(U) = 1001 \)[/tex]
- The size of set [tex]\( A \)[/tex]: [tex]\( n(A) = 60 \)[/tex]
- The size of set [tex]\( B \)[/tex]: [tex]\( n(B) = 40 \)[/tex]
- The size of the complement of the union of sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex]: [tex]\( n(\overline{A \cup B}) = 5 \)[/tex]

2. Calculate the Size of the Union [tex]\( A \cup B \)[/tex]:
The complement of [tex]\( A \cup B \)[/tex] consists of all elements that are not in [tex]\( A \cup B \)[/tex]. Therefore, the size of the union [tex]\( A \cup B \)[/tex] can be derived from the size of the universal set [tex]\( U \)[/tex] minus the size of its complement.
[tex]\[ n(A \cup B) = n(U) - n(\overline{A \cup B}) \][/tex]
Substitute the given values:
[tex]\[ n(A \cup B) = 1001 - 5 = 996 \][/tex]

3. Apply the Principle of Inclusion-Exclusion:
The principle of inclusion-exclusion for the union of two sets states that:
[tex]\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \][/tex]
Rearrange this equation to isolate [tex]\( n(A \cap B) \)[/tex]:
[tex]\[ n(A \cap B) = n(A) + n(B) - n(A \cup B) \][/tex]

4. Substitute the Known Values:
Substitute the calculated value for [tex]\( n(A \cup B) \)[/tex] and the given values for [tex]\( n(A) \)[/tex] and [tex]\( n(B) \)[/tex]:
[tex]\[ n(A \cap B) = 60 + 40 - 996 \][/tex]

5. Perform the Calculation:
[tex]\[ n(A \cap B) = 100 - 996 = -896 \][/tex]

Thus, the value of [tex]\( n(A \cap B) \)[/tex] is [tex]\(-896\)[/tex].
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