Question

Out of all the patients in a hospital 89% are found to be suffering from heart ailment and 98% are suffering from lungs infection. If K% of them are suffering from both ailments, then K can not belong to the set :

• A. \{84, 86, 88, 90\}
• B. \{80, 83, 86, 89\}
• C. \{79, 81, 83, 85\}
• D. \{84, 87, 90, 93\}

Hint:

For any intersection \(n(A \cap B)\) of subsets of a universal set \(U\), the bounds are:
\(n(A) + n(B) - n(U) \leq n(A \cap B) \leq \min(n(A), n(B))\).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This problem involves the Principle of Inclusion-Exclusion for two sets. Let \(H\) be the set of patients with heart ailment and \(L\) be the set of patients with lung infection. We need to find the range of the intersection \(H \cap L\).
Step 2: Key Formula or Approach:
For two sets \(H\) and \(L\):
\[ n(H \cup L) = n(H) + n(L) - n(H \cap L) \]
Since the total percentage cannot exceed 100%, \(n(H \cup L) \leq 100\).
Also, the intersection cannot exceed the size of the smaller set: \(n(H \cap L) \leq \min(n(H), n(L))\).
Step 3: Detailed Explanation:
Given: \(n(H) = 89\), \(n(L) = 98\), and \(n(H \cap L) = K\).
Substituting into the formula:
\[ n(H \cup L) = 89 + 98 - K = 187 - K \]
Applying the constraints:
1. \(n(H \cup L) \leq 100 \implies 187 - K \leq 100 \implies K \geq 87\).
2. \(n(H \cup L) \geq \max(n(H), n(L)) \implies 187 - K \geq 98 \implies K \leq 89\).
Thus, the valid range for \(K\) is \([87, 89]\).
Now, we check the options to see which set contains values that fall entirely outside this range.
(A) \{84, 86, 88, 90\} (Contains 88, which is in range)
(B) \{80, 83, 86, 89\} (Contains 89, which is in range)
(C) \{79, 81, 83, 85\} (None of these values are in \([87, 89]\))
(D) \{84, 87, 90, 93\} (Contains 87, which is in range)
Step 4: Final Answer:
Since none of the elements in the set \{79, 81, 83, 85\} satisfy the condition \(87 \leq K \leq 89\), \(K\) cannot belong to this set.