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

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.