A and B are two sets such that A has 12 elements, B has 17 elements and A È B contains 21 elements. Find the number of elements in A Ç B.

n (A) = 12, n (B) = 17 and n (A È B) = 21

Using the fact,

n ( A È B) = n (A) + n (B) - n ( A Ç B),

21 = 12 + 17 - n ( A Ç B )

\ n ( A Ç B ) = 29 - 21 = 8

In a class of 50 students, 35 students play foot ball, 25 students play both football as well as base ball. All the students play at least one of the two games. How many students play base ball ?

Let F be the set of the students who play foot ball, and C be the set of students who play base ball.

Then we have n (C È F) = 50

n (f) = 35 and n (C Ç F) = 25

Now the problem can be visualized by means of a Venn
diagram as in the adjoining figure.

Thus we have n ( C È F ) = n (F) + n (C) - n (C Ç F )

\ The number of students who play only baseball

     = n (C) - n ( C Ç F )

     = n (C È F) - n (F)

     = 50 - 35

     = 15

Example

If A Ì B , B Ì A and B Ì C, then find which of the sets A, B and C are equal.

Solution :

If A Ì B, B Ì C      \ A Ì C

But given that C Ì A      \ A = C . . . (1)

Now, A = C Þ A Ì B and C Ì B . . . (I)

But given that B Ì C . . . (II)

From (I) and (II) B = C . . . (2)

From (1) and (2) A = B = C

Index

2.1 Sets
2.2 Operations on Sets
2.3 The Algebra of Sets

Chapter 3