Question

In a School 150 students passed X Standard examination. 95 students applied for
Group I and 82 students applied for Group II in the Higher Secondary course. If 20
students applied neither of the two, how many students applied for both groups?

Answer 1

Let A = set of students who applied for group I
     B = set of students who applied for group II
     U = set of students who passed X standard exam
     A Λ B = set of students who applied for both groups
     A υ B = set of students who applied for group I or group II
    ~ (A υ B) = set of students who did not apply for any group

 U  = (A υ B) + ~(A υ B)

Cardinatlity of U = cardinality of ~(A υ B) + cardinality of (A υ B)
150 = 20 + cardinality (A υ B)

A υ B = 150-20 = 130
A υ B = A + B - A Λ B
Cardinality of (A υ B) = cardinality of A + cardinality of B - cardinality of (A Λ B)
                    130 = 95 + 82 - cardinality ( A Λ B)

Cardinality of A Λ B = 47

So 47 students applied for both groups.