## 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.

5.           Sam is a section chief for an electric utility company. The employees in his section cut down tall trees or climb poles. Sam recently reported the following information to the management of the utility.

Out of 100 employees in my section, 55 can cut tall trees, 50 can climb poles, 11 can do both, 6 can't do any of the two. Is this information correct?

Solution:

Given                 n(U)      =   100

n(T∩P)      =      11

n(T∪P)'     =        6

n(T)          =       55

n(P)          =       50 From the Venn diagram, we know that

n(U)        =    n(T∪P) + n(T∪P)'

=     [44+11+39] + 6

=       100

So the given information is correct.

6.          A and B are two sets such that n(A-B) = 32 +x, n(B-A) = 5x and  n(A∩B) = x.  Illustrate the information by means of a Venn diagram. Given that n(A) = n(B). Calculate (i)the value of x  (ii) n(A∪B).

Solution:

Given that

n(A-B)  = 32 +x

n(B-A) = 5x

n(A∩B) = x

and                      n(A) = n(B)

Some important results are

n(A)     =  n(A-B) + n(A∩B)

n(B)     =  n(B-A) + n(A∩B)

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

But given that n(A)  = n(B)

So,           n(A-B) + n(A∩B) =  n(B-A) + n(A∩B)

n(A-B)        =  n(B-A)

32+x         =     5x

5x-x          =     32

4x           =     32

(i)              x            =     32/4  = 8

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

=    32+x +   x       + 5x

=    32  + 7x

=     32  +  7(8)

=      88

7.            The following table shows the percentage of the students of a school who participated in Elocution and Drawing competitions. Draw a Venn diagram to represent this information and use it to find the percentage of the students who

(i)  participated in Elocution only

(ii) Participated in  Drawing only

(iii) Do not participate in any one of the competitions.

Solution: (i)  Participated in Elocution only  = 35

(ii) Participated in Drawing only   =  25

(ii) Do not participated in any one

of the competitions           =  100-(35+20+25)

=  100-80

=     20

8.           A village has total population 2500. Out of which 1300 use brand A soap and 1050 use brand B soap and 250 use both brands. Find the percentage of population who use neither of these soaps.

Solution:

Given       n(U)       =  2500

n(A)       =  1300

n(B)       =  1050

n(A∩B)    =   250

We know that n(A∪B)     =   n(A) + n(B) - n(A∩B)

=   1300+1050-250

=    2100

n(A∪B)'    =    n(U) - n(A∪B)

=    2500-2100

=      400

Percentage of population who use neither of these soaps

=     400/2500

=      0.16  x 100

=        16%

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.