Question

If \[n\left( U \right) = 60\], \[n\left( A \right) = 21\], \[n\left( B \right) = 43\] then greatest value of \[n\left( {A \cup B} \right)\] and least value of \[n\left( {A \cup B} \right)\] are:
A) 60,43
B) 50,36
C) 70,44
D) 60,38

Answer 1

Hint:
Here, we will find the union of two sets A and B and check whether it is less than, greater than or equal to the number of elements in the universal set U. If it is equal or less than the number of elements in the universal set U that will be the answer, if not then the number of elements in the universal set U will be the greatest value. For least value, we will check which of the two sets has the least number that will be the least value of the union of two sets.

Complete step by step solution:
As it is given to us, \[n\left( U \right) = 60,n\left( A \right) = 21,n\left( B \right) = 43\], this means universal set has 60 elements, set A have 21 elements and set B have 43 elements.
Now the union of two set will be,
\[n\left( {A \cup B} \right) = n\left( A \right) + n\left( B \right)\]
Substituting the value \[n\left( U \right) = 60,n\left( A \right) = 21\] and \[n\left( B \right) = 43\] in the above equation, we get
\[ \Rightarrow n\left( {A \cup B} \right) = 21 + 43\]
Adding the terms, we get
\[ \Rightarrow n\left( {A \cup B} \right) = 64\]
Now, as we can see that the element in union of two sets is greater than elements in the universal set
Therefore,
\[\begin{array}{l}n\left( {A \cup B} \right) = n\left( U \right)\\ \Rightarrow n{\left( {A \cup B} \right)_{\max }} = 60\end{array}\]
So the greatest value of \[n\left( {A \cup B} \right)\] is 60.
Next, we will find the least value of the union of two sets and as \[n\left( B \right) > n\left( A \right)\], therefore
\[{\left( {n\left( {A \cup B} \right)} \right)_{least}} = n\left( B \right)\]
\[ \Rightarrow {\left( {n\left( {A \cup B} \right)} \right)_{least}} = 43\]
So the least value of \[n\left( {A \cup B} \right)\] is 43.

Hence, option (A) is correct.

Note:
When two sets of a Universal set are disjoint the union of them yields the total number of elements in the universal set. If we add the elements, we are getting a number of elements more than that in the universal set. It means that the two sets have some elements which are common. So the greatest value of the union of two sets becomes equal to the number of elements in the universal set.