Question

Let A and B be 2 sets having elements 4 and 7 respectively. Then write the maximum number of elements that A∪B can have.

Answer 1

Hint: As we have asked to find the maximum number of elements in A∪B, we must know the formula of n (A∪B) which is equal to n (A) + n (B) – n (A∩B) where n represents number of elements. Now, n (A∪B) is maximum when n (A∩B) will be minimum and minimum of n (A∩B) is 0.

Complete step-by-step answer:
Let us assume that:

 Number of elements of A is equal to n (A).

 Number of elements of B is equal to n (B).

Number of elements of A∩B is equal to n (A∩B).

Now, the formula for the elements of A U B is as follows:

$n\left( A\cup B \right)=n\left( A \right)+n\left( B \right)-n\left( A\cap B \right)$

It is given in the question that, number of elements in set A is equal to 4 and number of elements in set B is equal to 7. So, we can write:

n (A) = 4, n (B) = 7

n (A∩B): the number of elements which are common in both the sets A and B.

We have to find the maximum number of elements which are in n(A∪B). So, the maximum of n(A∪B) is when n(A∩B) is minimum and the minimum of n(A∩B) is 0.

$\begin{align}

  & n\left( A\cup B \right)=n\left( A \right)+n\left( B \right)-n\left( A\cap B \right) \\

 & \Rightarrow n\left( A\cup B \right)=4+7-0 \\

 & \Rightarrow n\left( A\cup B \right)=11 \\

\end{align}$

Hence, the maximum number of elements in A∪B is 11.

Note: You might be wondering, why for maximizing the n (A∪B) we have to minimize n (A∩B) and why the minimum value of n (A∩B) is 0.

In the formula for n (A∪B),

$n\left( A\cup B \right)=n\left( A \right)+n\left( B \right)-n\left( A\cap B \right)$

L.H.S will be maximized when in the R.H.S; the term after minus sign will be minimized. Hence, n (A∩B) must be minimized.

The minimum possible value of n (A∩B) is 0 means that the elements in the set A and B are such that there is not a single element is common in both the sets. And it could be possible that we have two sets and the sets have no element in common.

When no elements is common in both the sets, the n (A∩B) is ∅.

The set which has no element in it is called a null set or an empty set.