Operation Of Set And Venn Diagrams

CONTENT: Use of Venn diagrams to solve problems involving three sets

Three Venn diagram:

Example:

A school has 37 vacancies for teachers, out of which 22 are for English, 20 for History and 17 for Fine Art. Of these vacancies 11 are for both English and History, 8 for both History and Fine Art and 7 for English and Fine Art. Using a Venn diagram, find the number of teachers who must be able to teach:

(a.)    all the three subjects

(b.)    Fine Art only

(c.)    English and History but not Fine Art.

Solution:

Let µ = {All vacancies for teachers}

      E = {English vacancies}

      H = {History vacancies}

      F = {Fine Art vacancies}

µ = 37, n(E)= 22, n(H)= 20, n(F)= 17, n{EnH}= 11, n(HnF)= 8, n(EnF)= 7

• Let n(EnFnH) = y

   n (EnH^InF)= n(E)- (7-y+y+11-y)

                    = 22- (18-y)            =   4 + y

   n(E^InHnF) = n(H) – (11-y+y+8-y)

                     = 20- (19-y)          =    1+y

    n(E^InH¹nF)= n(F) – ( 7-y +y+8-y)

                      = 17 – (15- y)        =   2 +y

µ= 4+y+11-y+1+y+y+8-y+7-y+2+y

37= 33 + y

y = 37- 33

y = 4.               

      n(EnHnF) = 4 teachers

(2.)    Fine Art only, n(E^InH^InF) = 2+ y

                                                      = 2+4    = 6 teachers

            (3.)    English and History but not Fine Art i.e English and History only

                         n(EnHnF^I) = 11-y

                                           = 11- 4 = 7 teachers.

General Evaluation

1. n(P) =4 means that these are 4 element in set P. given that n(XƲY)= 50, n(X)=20 and n(Y)= 40. Find       n(X∩Y)
2. find the sum of the first five terms of GP 2,6,18……..
3. the twelfth term of a linear sequence is 47 and the sum of the first three term is 12. Find the sum of the       first 15 terms of the sequence  
4. At a meeting of 35 teachers, the analysis of how Fanta, Coke and Pepsi were served as refreshments is as

follows. 15 drank Fanta, 6 drank both Fanta and coke, 18 drank Coke, 8 drank both Coke and Pepsi, 20 drank Pepsi, and 2 drank all the three types of drink. How many of the teachers drank I Coke only II Fanta and Pepsi but not Coke.

• Given n(XUY) = 50, n(X) = 20 and n(Y) = 40, determine n(XnY)

Reading Assignment: Read Sets, Further Mathematics Project II, page 1- 13.

Weekend Assignment

Theory

1. In a school of 300 students, 110 offered French, 110 Hausa language, 180 History, 40 French and Hausa, 50 Hausa and History, 60 French and History while 30 did not offer any of the three subjects.

1. Draw a Venn diagram to represent the data
2. Find the number of students who offered   I all the three subjects II History alone.

• In a certain class 22, pupils take one or more of chemistry, economic and government. 12 take economics (e), 8 take government (G) and 7 take chemistry (c). nobody takes economics and chemistry and 4 pupils takes economic and government
• Using set notation and the letters to indicate above, write down the two statements in the last sentence
• Draw the venn diagram to illustrate the information

c)  How many pupils take; (i.)  Both chemistry and government  (ii.) Government only