Question

There are \[200\] individuals with a skin disorder, \[120\] has been exposed to chemical \[C1\], \[50\] to chemical \[C2\], and \[30\] to both \[C1\] and \[C2\]. Find the number of individuals exposed to chemical \[C1\] or chemical \[C2\], chemical \[C1\] but not \[C2\], chemical \[C2\] but not chemical \[C1\]. Solve without using Venn diagrams.

Answer 1

Hint: In order to solve the question, we are going to consider two sets \[A\] and \[B\]. According to the information given, we have the values of the number of individuals in set A , that in set \[B\] and that for the set \[A\] intersection \[B\]. These values can be used to find the values for \[A\] union\[B\], \[A\] not \[B\] and \[B\] not \[A\].

Formula used: The formula used to find \[A\]union\[B\]
\[n\left( A\cup B \right)=n\left( A \right)+n\left( B \right)-n\left( A\cap B \right)\]
For \[A\]not\[B\]
\[n\left( A-B \right)=n\left( A \right)-n\left( A\cap B \right)\]
And for \[B\]not\[A\].
\[n\left( B-A \right)=n\left( B \right)-n\left( A\cap B \right)\]

Complete step-by-step solution:
Let us consider the following sets
Set \[A\] be the individuals exposed to chemical \[C1\]
Thus, \[n\left( A \right)=120\]
Set \[B\] be the individuals exposed to chemical \[C2\]
Thus, \[n\left( B \right)=50\]
As the individuals exposed to both the chemicals is \[30\]
So, \[n\left( A\cap B \right)=30\]
To find the individuals exposed to chemical \[C1\] or chemical \[C2\]
We use the formula
\[n\left( A\cup B \right)=n\left( A \right)+n\left( B \right)-n\left( A\cap B \right)\]
Putting the values from above, we get
\[
    \Rightarrow n\left( A\cup B \right)=120+50-30 \\
  \Rightarrow n\left( A\cup B \right)=140 \\
\]
The individuals exposed to chemical \[C1\] or chemical \[C2\] are \[140\]
To find the individuals exposed to chemical \[C1\] but not chemical \[C2\]
We use the formula
\[
    \Rightarrow n\left( B-A \right)=n\left( B \right)-n\left( A\cap B \right) \\
  \Rightarrow n\left( B-A \right)=50-30 \\
  \Rightarrow n\left( B-A \right)=20 \\
\]
To find the individuals exposed to chemical \[C2\] but not chemical \[C1\]
We use the formula
\[
    \Rightarrow n\left( A-B \right)=n\left( A \right)-n\left( A\cap B \right) \\
  \Rightarrow n\left( A-B \right)=120-3 \\
  \Rightarrow n\left( A-B \right)=90 \\
\]

Note: To find the number of individuals exposed to chemical \[C1\] and \[C2\], it is very important to deduct the number of individuals exposed to both, while for those who are exposed to \[C1\] not \[C2\] or vice versa, we can find by deducting the individuals exposed to both from total number exposed to \[C1\] or vice versa.