Question

If $ A $ and $ B $ are any two non-disjoint sets, then the value of $ n\left( {A \cup B} \right) $ is equal to:
(A) $ n\left( A \right) + n\left( B \right) $
(B) $ n\left( A \right) + n\left( B \right) - n\left( {A \cap B} \right) $
(C) $ n\left( A \right) + n\left( B \right) + n\left( {A \cap B} \right) $
(D) $ n\left( A \right) - n\left( B \right) $

Answer 1

Hint: In the given question, we are given two events A and B that are not disjoint. We must know the definition of disjoint sets in order to solve the questions. Disjoint sets are the sets that do not have anything in common. This means that there is no absolutely no element in the intersection of the events. So, we will find the value of the expression $ n\left( {A \cup B} \right) $ using the formula for the number of elements in the union of two sets.

Complete step-by-step answer:
In the given question, we have to find the number of elements in the union of two sets A and B when we are given that the sets are not disjoint.
So, the two sets may have some elements in common. Hence, we can infer that the intersection of these two sets A and B is not a null set.
Now, we know that we can add up the number of elements in two sets easily but that will result in double counting of the sets that are common in both the sets A and B. So, what we will do is first add the number of elements in both the sets A and B and then subtract the number of elements that are common to both the sets A and B so as to get the elements that are present in either of the two sets.
Hence, we get, $ n\left( {A \cup B} \right) = n\left( A \right) + n\left( B \right) - n\left( {A \cap B} \right) $ .
So, the correct answer is “Option B”.

Note: These problems involve the core concepts of sets. The formula derived in the questions can be learnt and remembered so that it can be applied directly in more such questions. We should handle the calculations with utmost care so as to be sure of the answer.