Question

If $n\left( U \right) = 60,n\left( A \right) = 21,n\left( B \right) = 43$ the greatest value of $n\left( {A \cup B} \right)$ and the 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: We are given that $n\left( U \right) = 60,n\left( A \right) = 21,n\left( B \right) = 43$and we know that $n\left( {A \cup B} \right)$represents the number of elements in the union of set A and set B. As the union consists of all the elements from both the sets it can never be less than the number of elements of the set A and B . Same way the number of elements in the union cannot exceed the number of elements in the universal set. From this we get the greatest and least value of $n\left( {A \cup B} \right)$.

Step by step solution :
We are given that the universal set consists of 60 elements and the set A contains 21 elements and set B contains 43 elements.
Now we need to discuss the number of elements in the union of the two sets. The union of the sets contain the elements of both the sets.
So $n\left( {A \cup B} \right)$ can never be less than the number of elements in set A or B
Since here set B has more elements than set A.
We get that the number of elements in the union cannot be less than the number of elements in set B , that is 43.
Therefore the least value of $n\left( {A \cup B} \right)$ is 43.
Same way the number of elements in the union of the sets can never be more than the number of elements in the universal set
So the greatest value of $n\left( {A \cup B} \right)$ is 60.

Hence the least and greatest value are 43 and 60.
Therefore the correct answer is option A.

Note :
1) In English, the union of two sets A and B is the set containing elements that are either in A or in B.
2) An element belongs to the intersection of two sets if the element is in both set A and in set B.
3) In mathematics, the cardinality of a set is a measure of the "number of elements" of the set.