Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R.
Assertion A: In a class of 40 students, 22 drink Sprite, 10 drink Sprite but not Pepsi. Then the number of students who drink both Sprite and Pepsi is 15.
Reason R: For any two finite sets A and \( \mathrm{B}, \, \mathrm{n} \mathrm{A} \) = \( \mathrm{n} \mathrm{A} - \mathrm{B} + \mathrm{n} \mathrm{A} \cap \mathrm{B} \).
In the light of the above statements, choose the most appropriate answer from the options given below.
Step-by-step Solution:
To solve this, we must evaluate the truthfulness of the Assertion (A) and the Reason (R) separately.
Assertion: In a class of 40 students, 22 drink Sprite, 10 drink Sprite but not Pepsi. Then the number of students who drink both Sprite and Pepsi is 15.
Let's verify this using set theory. Let S = Sprite drinkers and P = Pepsi drinkers.
The set of all Sprite drinkers is made up of those who only drink Sprite and those who drink both. So, the correct relationship is:
$n(S) = n(S - P) + n(S \cap P)$
Plugging in the given numbers:
$22 = 10 + n(S \cap P)$
$n(S \cap P) = 22 - 10 = 12$
The actual number of students who drink both is 12. The assertion claims the number is 15, which is incorrect. Therefore, Assertion (A) is false.
Reason: For any two finite sets A and B, $n(A) = n(A - B) + n(A \cap B)$.
This is a fundamental principle of set theory. It states that the total number of elements in a set (A) is the sum of the elements that are exclusively in that set ($A - B$) and the elements that are shared with another set ($A \cap B$). This statement is a correct mathematical formula. Therefore, Reason (R) is true.
Since Assertion (A) is false and Reason (R) is true, the correct option is D.