Question:
In a group of 50 persons, 14 drink tea but not coffee and 30 drink tea. Find:
(i) how may drink tea and coffee both;
(ii) how many drink coffee but not tea.
Solution:
Let A & B denote the sets of the persons who drink tea & coffee, respectively .
Given:
$n(A \cup B)=50$
$n(A)=30$
$n(A-B)=14$
(i) $n(A-B)=n(A)-n(A \cap B)$
$\Rightarrow 14=30-n(A \cap B)$
$\Rightarrow n(A \cap B)=16$
Thus, 16 persons drink tea and coffee both.
(ii) $n(A \cup B)=n(A)+n(B)-n(A \cap B)$
$\Rightarrow 50=30+n(B)-16$
$\Rightarrow n(B)=36$
We have to find $n(B-A)$
$\Rightarrow n(B-A)=n(B)-n(A \cap B)$
$\Rightarrow n(B-A)=36-16=20$
Thus, 20 persons drink coffee but not tea.