Question

Let A and B be the sets containing \[3\] and \[6\] elements respectively. Find the maximum and minimum numbers of elements in \[A \cup B\]?

Answer 1

Hint: According to the question, we will first find \[n(A)\& \,\,n(B)\]. Here, ‘n’ means the number of elements in a set. Set here means collection of elements in brackets. \[A \cup B\] means the union of set A and set B. This means that this set contains all the elements present in set A and in set B.

Complete step-by-step answer:
According to the question, the given part is that A has \[3\]elements, and B has \[6\]elements. So, we can say that:
\[n(A) = 3\,\,\& \,\,n(B) = 6\]
Here ‘n’ means the number of elements of the respective set.
Now, we will find the total number of elements in both A and B sets. This means that we can get the maximum numbers of elements in \[A \cup B\]. For that we have to add all the elements of both the sets. There is a formula for this:
\[n(A \cup B) = n(A) + n(B)\]
Here, we will put the values of \[n(A)\& \,\,n(B)\]
\[ \Rightarrow n(A \cup B) = 3 + 6\]
\[ \Rightarrow n(A \cup B) = 9\]
Now, we will calculate the minimum number of elements in both A and B sets which is \[A \cup B\]. We can say that if A is a subset of B, then \[A \cup B\]will be minimum.
\[ \Rightarrow n(A \cup B) = n(A)\]
\[ \Rightarrow n(A \cup B) = 6\]
Therefore, the maximum number of elements in \[A \cup B\]is \[9\].
The minimum number of elements in \[A \cup B\] is \[6\].

Note: According to the above method, the question gets solved very easily, but many students get confused at a place and they make a mistake. When they calculate the maximum number of elements, they add the elements of both the sets. But when they calculate the minimum elements, then they subtract the elements.