JEE Main 2021 — Sets And Relations Question with Solution

This solution begins by applying the principle of inclusion and exclusion, which in the context of this problem, is represented by the formula :

n(A ∪ B) ≥ n(A) + n(B) - n(A ∩ B)

Here, n(A ∪ B) represents the total number of patients in the hospital, which is 100%. n(A) represents the proportion of patients with a heart ailment (89%), and n(B) represents the proportion of patients with a lung infection (98%).

By rearranging this formula, the solution establishes an inequality for n(A ∩ B), the proportion of patients suffering from both ailments :

100% ≥ 89% + 98% - n(A ∩ B)

Therefore,

n(A ∩ B) ≥ 87%

Next, the solution notes that n(A ∩ B) cannot be greater than the smaller of n(A) and n(B), since it cannot be larger than the smallest group. Thus, we have another inequality :

n(A ∩ B) ≤ 89%

Combining these two inequalities gives :

87% ≤ n(A ∩ B) ≤ 89%

Hence, the proportion of patients suffering from both ailments must be a value between 87% and 89% inclusive. So, the set of values {79,81,83,85} which are all less than 87% are values that n(A ∩ B) cannot belong to. Therefore, option C is the correct answer.