Question

Let \[n\left( U \right) = 700\] , \[n\left( A \right) = 200\] , \[n\left( B \right) = 300\] and \[n\left( {A \cap B} \right) = 100\] , then \[n\left( {{A^c} \cap {B^c}} \right) = \]
\[\left( 1 \right){\text{ }}400\]
\[\left( 2 \right){\text{ 6}}00\]
\[\left( 3 \right){\text{ 3}}00\]
\[\left( 4 \right){\text{ 2}}00\]

Answer 1

Hint: In this question we will use the property of De Morgan’s Law. For this first we have to find the value of \[n\left( {A \cup B} \right)\] and then to find its complement we have to subtract this from the number of terms in the universal set. After finding the value of compliment of \[n\left( {A \cup B} \right)\] we will use the property and can get the value of \[n\left( {{A^c} \cap {B^c}} \right)\] .

Complete step-by-step answer:
It is given to us that number of elements in a universal set is \[700\] ,number of elements in set A is \[{\text{2}}00\] , number of elements in set B is \[300\] and the number of common elements between both the sets A and B is \[100\] . And we have to find \[n\left( {{A^c} \cap {B^c}} \right)\] .
Here in this question we will use De Morgan’s law which states that the complement of the union of two sets is the intersection of their complements and the complement of the intersection of two sets is the union of their complements. According to De Morgan’s law \[{\left( {A \cup B} \right)^c} = {A^c} \cap {B^c}\] and \[{\left( {A \cap B} \right)^c} = {A^c} \cup {B^c}\] .Now because we have to find the number of elements in \[{A^c} \cap {B^c}\] ,therefore we will use the property \[{\left( {A \cup B} \right)^c} = {A^c} \cap {B^c}\] .From this it is clear that to find \[n\left( {{A^c} \cap {B^c}} \right)\] first we have to find \[n\left( {A \cup B} \right)\] . Formula to find \[n\left( {A \cup B} \right)\] is given below
\[n\left( {A \cup B} \right) = n\left( A \right) + n\left( B \right) - n\left( {A \cap B} \right)\] ---------- (i)
Now put the values in equation (i) that are given in the question, so by this the above equation becomes
\[ \Rightarrow n\left( {A \cup B} \right) = 200 + 300 - 100\]
On doing addition and subtraction of numbers we get
\[ \Rightarrow n\left( {A \cup B} \right) = 400\]
Therefore, the required value of \[n\left( {A \cup B} \right)\] is \[400\] .
Now we have to find the complement of \[n\left( {A \cup B} \right)\] .To find its complement the formula given below is used
\[n{\left( {A \cup B} \right)^c} = n\left( U \right) - n\left( {A \cup B} \right)\]
As it is given that \[n\left( U \right) = 700\] and the required value of \[n\left( {A \cup B} \right) = 400\] .So, substitute these values in the above written equation.
\[n{\left( {A \cup B} \right)^c} = 700 - 400\]
\[ \Rightarrow n{\left( {A \cup B} \right)^c} = 300\]
As we know that \[{\left( {A \cup B} \right)^c} = {A^c} \cap {B^c}\] .Therefore we can say that \[{A^c} \cap {B^c} = 300\] .
Hence the correct option is \[\left( 3 \right){\text{ 3}}00\]
So, the correct answer is “Option 3”.

Note: Check all the given conditions and think about the procedure and how you can do this question. Remember the property comes under De Morgan’s Law because this is the easiest way by which you can solve this question. Don’t get confused by looking at the question, it’s just based on a single property. Avoid making mistakes during calculations.