Question

Let A, B, C be finite sets. Suppose that $n(A)=10, n(B)=15, n$ $(C)=20, n(A \cap B)=8$ and $n(B \cap C)=9$. Then the possible value of $n ( A \cup B \cup C )$ is

• A. 26
• B. 27
• C. 28
• D. Any of the three values 26, 27, 28 is possible

Answer 1

Answer: (D)

Any of the three values 26, 27, 28 is possible

Answer 2

$n(A)=10, n(B)=15, n(20), n(A \cap B)=8, n(B \cap C)=9$
possible value of $n$ (AUBUC)
$n(A \cup B \cup C)=n(A)+n(B)+n(C)-n(A \cap B) -n(B \cap C)-n(A \cap C) +n(A \cap B \cap C)$
$=10+15+20-8-9-n(A \cap C)+n(A \cap B \cap C)$
$=28-n(A \cap C)+n(A \cap B \cap C)$
We see here,
$28-n(A \cap C)+n(A \cap B \cap C) \geqslant 0$
$\Rightarrow n(A \cup B \cup C) \leqslant 28\,\,\,...(1)$
We see, $n(A \cup B)=n(A)+n(B)-n(A \cap B)$
$=10+15-8$
$=17$
$n(B \cup C) =n(B)+n(C)-n(B \cap C)$
$=15+20-9$
$=26$
Obviously, $n(A \cup B \cup C) \geqslant 26$ also,
$n(A \cup B \cup C) \geqslant 17\,\,\, ...(2)$
$\therefore$ from (1) & (2)
$26 \leqslant n(A \cup B \cup C) \leqslant 28$