Question

 Find ‘\[n\left( {{A^c} \cap {B^c}} \right) = \]’, given that ‘\[n\left( U \right) = 700\]’ \[n\left( A \right) = 200\]’ \[n\left( B \right) = 300\]’ \[n\left( {A \cap B} \right) = 100\]’.
A. \[400\]
B. \[600\]
C. \[300\]
D. \[200\]

Answer 1

Hint:
Recall De Morgan’s law.

Formula Used:
De Morgan’s Law:
\[{A^c} \cap {B^c} = {\left( {A \cup B} \right)^c}\]
Complementary formula:
\[n\left( {{A^c}} \right) = n\left( U \right) - n\left( A \right)\]
The formula of the number of element of \[A \cup B\]:
\[n\left( {A \cup B} \right) = n\left( A \right) + n\left( B \right) - n\left( {A \cap B} \right)\]




Complete step-by-step answer:
We have to find the value of \[n\left( {{A^c} \cap {B^c}} \right)\]
Apply De Morgan’s Law.
\[{A^c} \cap {B^c} = {\left( {A \cup B} \right)^c}\]
Calculating the number of elements of the above relation
\[n\left( {{A^c} \cap {B^c}} \right) = n{\left( {A \cup B} \right)^c}\]
Applying complementary formula on RHS
\[n\left( {{A^c} \cap {B^c}} \right) = n\left( U \right) - n\left( {A \cup B} \right)\]
\[n\left( {{A^c} \cap {B^c}} \right) = n\left( U \right) - \left[ {n\left( A \right) + n\left( B \right) - n\left( {A \cap B} \right)} \right]\]
Simplify the above equation:
\[n\left( {{A^c} \cap {B^c}} \right) = n\left( U \right) - n\left( A \right) - n\left( B \right) + n\left( {A \cap B} \right)\]
Substitute \[n\left( U \right) = 700\], \[n\left( A \right) = 300\], \[n\left( B \right) = 200\] and \[n\left( {A \cap B} \right) = 100\] in the above equation, we get;
\[n\left( {{A^c} \cap {B^c}} \right) = 700 - 300 - 200 + 100\]
\[n\left( {{A^c} \cap {B^c}} \right) = 300\]
The correct answer is option C.



Note:
Students often make mistakes when they apply De Morgan’s formula. They used a wrong formula that is \[\left( {{A^c} \cap {B^c}} \right) = {\left( {A \cap B} \right)^c}\]. The correct formula is \[\left( {{A^c} \cap {B^c}} \right) = {\left( {A \cup B} \right)^c}\].