Solutions to set-VII

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

 1.          In an examination 150 students secured first class in English or Maths. Out of these 50 students obtained first class in both English and Maths.  115 students secured first class in Maths.  How many students secured first class in English only?

Solution:            Given 

                     n(E∪M)           =   150

                     n(E∩M)            =    50

                           n(M)                =   115

               Now let us draw the Venn diagram for the above information. 

              First we have to enter 50 in E∩M part, Now we have n(M) = 115.  

So in M circle we have to enter only (115-50=) 65.

             We know 

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

                         150     = n(E) + 115 - 50

                     150    = n(E) +  65

                    n(E)     =  150-65  = 85           

        From the above Venn diagram 

                     150  =  65 + 50 + x

                       x   =   150-115

                           =      35

Number of students who secured first class in English only is 35

2.           In a group of 30 persons, 10 take tea but not coffee. 18 take tea. Find how many take coffee but not tea, if each person takes at least one of the drinks?

Solution:

Given in a group of 30 persons, each one takes at least one of the drinks.

So         n(T∪C)          =    30 

               n(T)                =    18

Also given that number of persons drink only tea is 10.

There fore, number of persons drink both tea and coffee 

                  n(T∩C) = n(T)-10    = 18-10   =8

          From the above Venn diagram,

                    30     =     10 + 8 +x

                      x     =      30 - 18

                             =       12

     Number of persons drink only coffee not tea is  12.


3.           In a village there are 60 families.  Out of these 28 families speak only one language and 20 families speak other language. How many families speak both the languages?

Solution:

Given          n(A∪B)    =  60

                      n(A)         =  28

                      n(B)         =  20

 To find n(A∩B) let us use the result

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

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

                                      =      60   - [28+20]

                                      =       60  -   48

                                      =           12

Number of families speak both the languages are  12


4.           In a school 150 students passed X standard Examination. 95 students applied for group I and 82 students applied for group II for the further studies. If 20 students applied neither of two, how many students applied for both groups?

Solution:

Given            n(U)       =    150

                    n(A)       =     95

                    n(B)       =     82

                    n(A∪B)'    =       20

  So                   n(A∪B)     =   n(U) - n(A∪B)'

                                          =   150  -   20

                                          =       130

                 From the above Venn diagram 

                   (95-x) + x + (82-x)  = 130

                                x               = (95+82) - 130

                                                =  177 -    130

                                                =       47

        Number of students applied for both the group  = 47





Parents and teachers can encourage the students to do the problems on their own and become master in union and intersection of sets.  If you have any doubt you can verify the solutions given in the above page 'Solutions to set-VII'. Still if you have any doubt you can contact us through mail, and we will help you to clear all your doubts. 

                                      Set Theory

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