Question:
In a group of 950 persons, 750 can speak Hindi and 460 can speak English. Find:
(i) how many can speak both Hindi and English:
(ii) how many can speak Hindi only;
(iii) how many can speak English only.
Solution:
Let A & B denote the sets of the persons who like Hindi & English, respectively.
Given:
$n(A)=750$
$n(B)=460$
$n(A \cup B)=950$
(i) We know:
$n(A \cup B)=n(A)+n(B)-n(A \cap B)$
$\Rightarrow 950=750+460-n(A \cap B)$
$\Rightarrow n(A \cap B)=260$
Thus, 260 persons can speak both Hindi and English.
(ii) $n(A-B)=n(A)-n(A \cap B)$
$n(A-B)=750-260=490$
Thus, 490 persons can speak only Hindi.
(iii) $n(B-A)=n(B)-n(A \cap B)$
$\Rightarrow n(B-A)=460-260$
$=200$
Thus, 200 persons can speak only English.