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Question: Let A and B be two sets such that \(n\left( A \right) = 52\),\(n\left( B \right) = 60\) and\(n\left(...

Let A and B be two sets such that n(A)=52n\left( A \right) = 52,n(B)=60n\left( B \right) = 60 andn(A∩B)=16n\left( {A \cap B} \right) = 16. Draw a Venn diagram and find: (i)n(A∪B)n\left( {A \cup B} \right) (ii)n(A−B)n\left( {A - B} \right) (iii)n(B−A)n\left( {B - A} \right)

Explanation

Solution

We will sketch the Venn diagram by the given data and we will solve the parts one-by-one as: (i) we will use the formula: n(A∪B)=n(A)+n(B)−n(A∩B)n\left( {A \cup B} \right) = n\left( A \right) + n\left( B \right) - n\left( {A \cap B} \right) , (ii) we will use the formula: n(A−B)=n(A)−n(A∩B)n\left( {A - B} \right) = n\left( A \right) - n\left( {A \cap B} \right), and (iii) we will use the formula: n(B−A)=n(B)−n(A∩B)n\left( {B - A} \right) = n\left( B \right) - n\left( {A \cap B} \right).

Complete step-by-step answer:
We are given two sets A and B.
We are required to draw a Venn diagram based on the given data: n(A)=52n\left( A \right) = 52,n(B)=60n\left( B \right) = 60 andn(A∩B)=16n\left( {A \cap B} \right) = 16
Venn diagram:

We are required to calculate the values of:
n(A∪B)n\left( {A \cup B} \right)
We will use the formula: n(A∪B)=n(A)+n(B)−n(A∩B)n\left( {A \cup B} \right) = n\left( A \right) + n\left( B \right) - n\left( {A \cap B} \right) to calculate the value of n(A∪B)n\left( {A \cup B} \right).
Substituting the values, we get
⇒n(A∪B)=52+60−16=96\Rightarrow n\left( {A \cup B} \right) = 52 + 60 - 16 = 96
Therefore, the value of n(A∪B)n\left( {A \cup B} \right)= 96.
n(A−B)n\left( {A - B} \right)
We will use the formula: n(A−B)=n(A)−n(A∩B)n\left( {A - B} \right) = n\left( A \right) - n\left( {A \cap B} \right) to calculate the value of n(A−B)n\left( {A - B} \right)
Substituting the values, we get
⇒n(A−B)=52−16=36\Rightarrow n\left( {A - B} \right) = 52 - 16 = 36
Therefore, the value of n(A−B)n\left( {A - B} \right)is 36.
n(B−A)n\left( {B - A} \right)
We will use the formula: n(B−A)=n(B)−n(A∩B)n\left( {B - A} \right) = n\left( B \right) - n\left( {A \cap B} \right)to calculate the value of n(B−A)n\left( {B - A} \right)
Substituting the values, we get
⇒n(B−A)=60−16=44\Rightarrow n\left( {B - A} \right) = 60 - 16 = 44
Therefore, the value of n(B−A)n\left( {B - A} \right)is 44.

Note: In this question, you may get confused at many places where we have used the formulae to calculate the parts of this question. The formulae used are properties of union and intersection of the sets which hold universally.