Week 5 – SS1 First Term Further Mathematics Notes

 
 WEEK FIVE
TOPIC: OPERATION OF SET AND VENN DIAGRAMS
CONTENT: Use of Venn diagrams to solve problems involving three sets

 Three Venn diagram:

  A        2    6    B
    5

     1    4
     3
    7
8 C

 
 
  1= AnBnC            5 = AnB^InC^I
2 = AnBnC^I        6 = A¹nBnC¹
3 = AnB^InC        7 = A¹nB^InC
4 = A^InBnC        8 = (AnBnC)¹
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

1. 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.

 
 Examples:

1. In a survey of 290 newspaper readers, 181 of them read daily times, 142 read the Guardian, 117 read the Punch and each read at least one of the paper, if 75 read the Daily Times and the Guardian,60 read the Daily Times and Punch and 54 read the Guardian and the punch
2. Draw a venn diagram to illustrate the information
3. How many read:
   1. all the three papers
   2. exactly two of the papers
   3. exactly one of the paper

 Solution
    n (E)= 290

 
 
 
 
 
 
 n(E)= 290
n(D)= 181
n(G)= 142
n(D∩G)= 75
n(D∩P)= 60
n(G∩P)= 54
from the venn diagram, readers who read Daily Times only
=181 – (160 – X + 75 – X +X)
=181 – (135 – X)
= 46 + X
Punch readers only
=117 – (60 – X + 54 – X + X)
117 – (114 – X)
117 – 114 + X
=3 +X
Guardian readers only
=142 – (75 – X + 54 – X + X)
=142 – (129 – X)
=142 – 129 + X
=13 + X
Where:
X is the number of readers who read all the three papers
Since the sum of the number of elements in all regions is equal to the total number of elements in the universal set, then:
46 + X + 75 – X + 13 + X + 60 – X + X + 54 – X + 3 + X = 90
251 + X = 290
X = 290 – 251
X= 39

 B(i): number of people who read all the three paper= 39
(ii) from the venn diagram, number of people who read exactly two papers
= 60 – X + 75 – X + 54 – X
=189 – 3X = 189 – 3(39) from the above
=189 – 117 = 72
(iii) also, from the venn diagram, number of people who read exactly only one of the papers
=46 + X + 13 + X + 3 +X
=162 +3X =162 + 3(39)
=162 + 117 = 179
(iv)number of Guardian reader only
=13 + X
=13 + 39 = 52

1. A group of students were asked whether they like History, Science or Geography. There responds are as follow

1. Represent the information in a Venn diagram
2. How many students were in the group? c) How many students like exactly two subjects

    Solution
   

3. n(E)= ?

   

    
    
    
    
    
    
    

1. Number of students in the group = sum of the elements in all the regions i.e

   Number of students in the group = 20 + 18 + 16 + 4 + 3 + 2 + 7 + 3 = 73

2. Number of students who like exactly two subject = 4 + 3 + 2= 9

 Evaluation

1. In a community of 160 people, 70 have cars ,82 have motorcycles, and 88 have bicycles, 20 have both cars and motorcycles,25 have both cars and bicycles, while 42 have both motorcycles and bicycles each person rode on at least any of the vehicles
2. Draw a venn diagram to illustrate the information
3. Find the number of people that has both cars and bicycles
4. How many people have either one of the three vehicles?

 
 
 
 
  N(U)

 
 
 
 
 
 The score of 144 candidates who registered for mathematics, physics and chemistry in an examination in a town are represented in the venn diagram above.

1. How many candidate register for both mathematics and physics?
2. How many candidate register for both mathematics and physics only?

 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.

5. 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
   

   1. In a class of 50 pupils, 24 like oranges, 23 like apples and 7 like the two fruits. How many students do not like oranges and apples? (a)7 (b) 6 (c) 10 (d)15
   2. In a survey of 55 pupils in a certain private schools, 34 like biscuits, 26 like sweets and 5 of them like none. How many pupils like both biscuits and sweet? (a) 5(b) 7 (c)9 (d)10
   3. In a class of 40 students, 25 speak Hausa, 16 speak Igbo, 21 speak Yoruba and each of the students speak at least of the three languages.

      If 8 speak Hausa and Igbo. 11 speak Hausa and Yoruba.6 speak Igbo and Yoruba. How many students speak the three languages? (a) 3 (b) 4 (c) 5 (d) 6

   Use the information to answer question 4 and 5
   N(U)=61
   

    
    

    
    
    
    
    
    The venn diagram above shows the food items purchased by 85 people that visit a store one week. Food items purchased from the store were rice, beans and gari.

   1. How many of them purchased gari only? (a)8 (b)10 (c) 14 (d)12
   2. How many of them purchased the three food items? (a) 5 (b)7 (c) 9 (d)11

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

    

   1. 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
   2. Using set notation and the letters to indicate above, write down the two statements in the last sentence
   3. Draw the venn diagram to illustrate the information

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