First slide

Operations on Sets

Question

From 50 students taking examinations in Mathematics, Physics and Chemistry, each of the students has passed in at least one of the subject. 37 passed Mathematics, 24 Physics and 43 Chemistry. At most 19 passed Mathematics and Physics, at most 29 Mathematics and Chemistry and at most 20 Physics and Chemistry. Then the largest possible number that could have passed all three examinations is

Easy

A

16
B

14
C

18
D

15

Solution

The given conditions can be expressed as
$n (M \cup P \cup C) = 50, n (M) = 37, n (p) = 24,$
$n (C) = 43, n (M \cap P) \leq 19, n (M \cap C) \leq 29$
and $n (P \cap C) \leq 20. n (M \cup P \cup C) =$
$n (M) + n (P) + n (C) - n (M \cap P)$
$- n (M \cap C) - n (P \cap C) + n (M \cap P \cap C)$
$\Rightarrow n (M \cap P \cap C) \leq n (M \cap P) + n (M \cap C)$
$+ n (P \cap C) - 54$ Therefore, the number of students is at most 19 + 29 + 20 - 54 = 14

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