## Solutions to set-VI

In this page, 'Solutions to set-VI' we are discussing how to do the problems given in problems on set-VI.

1.  Place the elements of the following sets in the proper location on the given Venn diagram.

U  =  {5,6,7,8,9,10,11,12,13}

M  =  {5,8,10,11}

N  =   {5,6,7,9,10}.

Some Important rules

For any two finite set A and B, we have the following rules.

1. n(A)         = n(A-B) + n(A∩B)
2. n(B)            =  n(B-A)  +  n(A∩B)
3. n(A∪B)       =  n(A-B)  +  n(A∩B)  +  n(B-A)
4. n(A∪B)       =  n(A)   +   n(B)  -  n(A∩B)
5. n((A∪B)      =  n(A)   +   n(B), when A∩B =
6. n(A) + n(A') =  n(U)

Following problems are based on the above rules.

2.    If A and B are two sets such that A has 50 elements, B has 65 elements and AB has 100 elements, how many elements in A∩B?

Solution:

Given n(A) = 50, n(B) = 65, n(AB) = 100.

By the rule

n(A∪B)       =  n(A)   +   n(B)  -  n(A∩B)

n(A∩B)       =  n(A)   +   n(B)  -  n(AB)

=    50    +     65  -     100

=          115        -       100

=                     15

3.     If A and B are two sets containing 13 and 16 elements respectively, then find the minimum and maximum number of elements in AB?

Solution:

n(A) = 13 and n(B) =16.

n( AB) must be either the elements of the bigger set, that is B or the addition of number of elements in both A and B.

If A is the subset of B, then  AB is the set B itself. Then the number of

AB is number of B itself. That is the minimum number of  AB.

So minimum of  AB is 16.

If A and B are two disjoint sets, then number of elements in  AB is the total number of elements in both A and B.

So maximum of  AB is 13+16 = 29.

4.     If n( A∩B) = 5, n(AB) = 35, n(A) = 13, find n(B)?

Solution:

By the rule

n(A∪B)       =  n(A)   +   n(B)  -  n(A∩B)

n(B)         = n(AB)+n(A∩B)-n(A)

=      35    +   5      -  13

n(B)         =           27

5.     If n(A) = 26, n(B) = 10, n(A∪B) = 30, n(A') =17, find n(A∩B) and n(U)?

Solution:

By the rule

n(A∪B)       =  n(A)   +   n(B)  -  n(A∩B)

n(A∩B)       =  n(A)   +   n(B)  -  n(AB)

=   26     +    10    -    30

n(A∩B)        =            6

By the rule

n(A) + n(A') =  n(U)

n(U)         =    26  +  17

=          43.

6.    If n(U) = 38, n(A) = 16,  n(A∩B) = 12, n(B') = 20, find n(A∪B)?

Solution:

By the rule

n(A) + n(A') =  n(U)

n(B)     = n(U)  -  n(B')

=   38   -    20

=       18

By the rule

n(A∪B)       =  n(A)   +   n(B)  -  n(A∩B)

=   16   +   18  -   12

=         34     -     12

=            22

7.   Let A and B be two finite sets such that n(A-B) = 30, n(A∪B) = 180, n(A∩B)= 60, find n(B)?

Solution:

By the rule