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