Question

In a survey of a population of 600 students, it is found that 250 can speak English (E), 210 can speak Hindi (H) and 120 can speak Tamil (T). If 100 people can speak both E and H, 80 can speak both E and T, 5 can speak both H and T and 20 can speak all three languages. Find the number of people who can speak E but not H and T. Find also the number of people who can speak neither E nor H nor T.

Answer 1

Hint: We solve this problem simply by using the set theory and its properties. We consider the total number of students as a universal set ‘U’ in which it is divided into three intersecting sets ‘E’, ‘H’, ‘T’. In the set theory ‘and’ refers to union and ‘or’ refers to the intersection.
By using these properties we calculate the number of students who speak E but not H and T as
\[\Rightarrow E\cap {H}'\cap {T}'\]
Then we calculate the number of people who can speak neither E nor H nor T as
\[\Rightarrow {E}'\cap {H}'\cap {T}'\]

Complete step-by-step solution:
Let us assume that the total number of students as the universal set ‘U’
So we can write
\[\Rightarrow n\left( U \right)=600\]
We are given that 250 can speak English (E) so we can write
\[\Rightarrow n\left( E \right)=250\]
Similarly, we are given that 210 can speak Hindi (H) and 120 can speak Tamil (T).
By converting all the given data to sets we get
\[\begin{align}
  & \Rightarrow n\left( H \right)=210 \\
 & \Rightarrow n\left( T \right)=120 \\
\end{align}\]
We are given that the combinations as 100 people can speak both E and H, 80 can speak both E and T, 5 can speak both H and T and 20 can speak all three languages.
By converting them to sets we get
\[\begin{align}
  & \Rightarrow n\left( E\cap H \right)=100 \\
 & \Rightarrow n\left( E\cap T \right)=80 \\
 & \Rightarrow n\left( H\cap T \right)=5 \\
 & \Rightarrow n\left( E\cap H\cap T \right)=20 \\
\end{align}\]
Now, let us find the first part.
(i) Number of students who can speak E but not H and T
We know that the number of students who can speak E but not H and T is given as
\[\Rightarrow n\left( E\cap {H}'\cap {T}' \right)\]
We know that \[n\left( {H}'\cap {T}' \right)=n\left( {{\left( H\cup T \right)}^{' }} \right)\]
By using this condition to above equation we get
\[\begin{align}
& \Rightarrow n\left( E\cap {H}'\cap {T}' \right)=n\left( E\cap {{\left( H\cup T \right)}^{' }} \right) \\
& \Rightarrow n\left( E\cap {H}'\cap {T}' \right)=n\left( E \right)-n\left( E\cap \left( H\cup T \right) \right) \\
& \Rightarrow n\left( E\cap {H}'\cap {T}' \right)=n\left( E \right)-n\left\{ \left( E\cap H \right)\cup \left( E\cap T \right) \right\} \\
\end{align}\]
We know that the union of two sets can be written as
\[\Rightarrow n\left( A\cup B \right)=n\left( A \right)+n\left( B \right)-n\left( A\cap B \right)\]
By using this condition to above equation we get
\[\begin{align}
& \Rightarrow n\left( E\cap {H}'\cap {T}' \right)=n\left( E \right)-[n(E\cap H)+n(E\cap T)-n\{(E\cap H)\cap (E\cap T)\}] \\
& \Rightarrow n\left( E\cap {H}'\cap {T}' \right)=n(E)-n(E\cap H)-n(E\cap T)+n(E\cap H\cap T) \\
\end{align}\]
Now, by substituting the required values in the above equation we get
\[\begin{align}
  & \Rightarrow n\left( E\cap {H}'\cap {T}' \right)=250-100-80+20 \\
 & \Rightarrow n\left( E\cap {H}'\cap {T}' \right)=90 \\
\end{align}\]
Therefore, we can say that there are 90 students who can speak English but not Hindi and Tamil.
Now, let us go for the second part
(ii) Number of people who can speak neither E nor H nor T.
We know that the number of students who speak neither E nor H nor T can be converted to set theory as
\[\Rightarrow n\left( {E}'\cap {H}'\cap {T}' \right)\]
By converting the intersection part to union part we get
\[\begin{align}
& \Rightarrow n(E’ \cap H’ \cap T’ )=n(E\cup H\cup T)’ \\
& \Rightarrow n(E' \cap H' \cap T' )=n(U)-n(E\cup H\cup T).........equation(i) \\
\end{align}\]
We know that the union of three sets is given as
\[\Rightarrow n(E\cup H\cup T)=n(E)+n(H)+n(T)-n(E\cap H)-n(H\cap T)-n(E\cap T)+n(E\cap H\cap T)\]
By substituting the required values in above equation we get
\[\begin{align}
  & \Rightarrow n(E\cup H\cup T)=250+210+120-100-80-5+20 \\
 & \Rightarrow n(E\cup H\cup T)=415 \\
\end{align}\]
Now by substituting the required values in the equation (i) we get
\[\begin{align}
  & \Rightarrow n(E' \cap H' \cap T' )=600-415 \\
 & \Rightarrow n(E' \cap H' \cap T' )=185 \\
\end{align}\]
Therefore, we can say that there are 185 people who speak neither English nor Hindi nor Tamil.

Note: Students may make mistakes in the transformation of set formulas.
The general formulas we used in solving this problem are
(i) \[n\left( {H}'\cap {T}' \right)=n\left( {{\left( H\cup T \right)}^{' }} \right)\]
(ii) \[n\left( A\cup B \right)=n\left( A \right)+n\left( B \right)-n\left( A\cap B \right)\]
(iii) \[n(E\cup H\cup T)=n(E)+n(H)+n(T)-n(E\cap H)-n(H\cap T)-n(E\cap T)+n(E\cap H\cap T)\]
These are simple standard results in set theory. As these are long formulas students may make mistakes in taking the formula which results in the wrong answer.
These points need to be taken care of.