Question

If a set \[A\] contains 3 elements and another set \[B\] contains 6 elements, then what is the minimum number of elements that \[A \cup B\] can have?
A.3
B.6
C.8
D.9

Answer 1

Hint: Here we need to find the number of elements present in the union of two sets. Here we will use the set formula to solve the problem. According to the set formula, the number of the elements present in the elements present in the union of two sets is equal to the addition of the number of elements present in the first set and the second set minus the number of elements present in the intersection of two sets.

Complete step-by-step answer:
Here we need to find the minimum number of elements present in the union of the two sets i.e. \[n\left( {A \cup B} \right)\]
It is given that:
\[\begin{array}{l}n\left( A \right) = 3\\n\left( B \right) = 6\end{array}\]
Here we will use the formula of the set to solve this problem.
According to the formula of set,
\[n\left( {A \cup B} \right) = n\left( A \right) + n\left( B \right) - n\left( {A \cap B} \right)\]
Now, \[n\left( {A \cup B} \right)\] is minimum when \[n\left( {A \cup B} \right)\]is maximum. This is possible only if \[A\] is subset of \[B\], i.e.,
\[ \Rightarrow n\left( {A \cap B} \right) = n\left( A \right) = 3\]
Now, we will substitute all the values here.
\[ \Rightarrow n\left( {A \cup B} \right) = 3 + 6 - 3\]
On adding and subtracting the numbers, we get
\[ \Rightarrow n\left( {A \cup B} \right) = 6\]
Hence, the minimum number of elements that can be present in the set \[A \cup B\] is equal to 6.
Hence, the correct option is option B.

Note: Here we have used the sets formula. We need to know the meaning and the term ‘set’. Set theory is defined as a branch of mathematics which deals with the properties of well-defined collections of objects which are also known as elements of the sets. Here elements may be numbers or the words. Generally, set theory deals with sets of elements and their characteristics as well as operations and relations of those sets.