Lesson Notes By Weeks and Term - Senior Secondary School 1

SUBJECT: MATHEMATICS

CLASS:  SS 1

DATE:

TERM: 2nd TERM

REFERENCE BOOK

• New General Mathematics SSS 1 M.F. Macrae et al 
• WABP Essential Mathematics For Senior Secondary Schools 1 A.J.S Oluwasanmi

 
WEEK FIVE           

TOPIC: SETS

CONTENT

• Venn Diagram and Venn diagram Representation.
• Using of Venn Diagram to Solve Problems Involving Two Sets.
• Using Venn Diagram to Solve Problems Involving Three Sets.

THE VENN DIAGRAM

The venn diagram is a geometric representation of sets using diagrams which shows different relationship between sets

Venn diagram representation

                              E or U  

The rectangle represents the universal set i.e E or U

             A

                           The oval shape represents the subset A.

The shaded portion represents the complement of set P i.e P| or Pc

The shaded portion shows the elements common to A and B i.e A∩B or A intersection B. 

The shades portion shows P intersection Q|i.e P∩Q|

The shaded portion shows AƲ B i.e A union B

                                                         U or E

This shows that P and Q have no common element. i.e P and Q are disjoint sets i.e P∩Q= Ф

          Q

      P

P is a subset of Q i.e P ⊂ Q

            U

    P                Q

PI∩ QI or (P Ʋ Q)|I. This shows elements that are neither in P nor Q but are represented in the universal set.

            R

This shows the elements common to set P,Q and R i.e the intersection of three sets P,Q and R i.e P∩Q∩R

This shows the elements in P only, but not in Q and R i.e P∩Q|∩R|

This shaded region shows the union of the three sets i.e PƲQƲ R

USING THE VENN DIAGRAM TO SOLVE PROBLEMS INVOVING TWO SETS

Examples:

1. Out of 400 final year students in a secondary school, 300 are offering Biology and 190 are offering Chemistry. If only 70 students are offering neither Biology nor Chemistry. How many students are offering (i)  both Biology and Chemistry? (ii) At least one of Biology or Chemistry?

Solution

            n(E)= 400

Let the number of students who offered both Biology and Chemistry be X i.e (B∩C)= X. from the information given in the question

n(E)= 400

n(B)= 300

n(C)= 190

n(BƲC)|=  70

Since the sum of the number of elements in all region is equal to the total number of elements in the universal set, then:

300 - x  + x +190 – x + 70 =400

560 – x= 400

-x= 400 – 560

X= 160

Number of students offering both Biology and Chemistry= 160

(ii)Number of students offering at least one of Biology and Chemistry from the Venn diagram  includes those who offered biology only, chemistry only and those whose offered both i.e

300 – x + 190 – x + x= 490 - x

490 – 160 (from (i) above) = 330

1. In a youth club with 94 members, 60 likes modern music and 50 likes traditional music. The number of them who like both traditional and modern music are three times those who do not like any type of music. How many members like only one type of music

Solution

Let the members who do not like any type of music = X

Then,

n(T n M)= 3X

Also,

n(E)= 94

n(M)=60

n(T)= 50

n(M u T)|= X

        n(E)= 94

    M                T

    60 – 3x3x   50 – 3x

                 X

Since the sum of the number of elements in all regions is equal to the total number of elements in the universal set, then

60 – 3X + 3X + 50 – 3X + X = 94

110 – 2X= 94

16= 2X

Divide both sides by 2

16= 2X

2     2

X= 8

Therefore number of members who like only one type of music are those who like modern music only  + those who like traditional music only.

60 -3x  + 50 – 3x

110 – 6x

= 110 – 6(8) = 110 - 48

= 62

EVALUATION

1. Two questions A and B were given to 50 students as class work.23 of them could answer question A but not B.  15 of them could answer B but not A. If 2x of them could answer none of the two questions and 2 could answer both questions.

1. Represent the information in a Venn diagram.    
2. Find the value of x

1. In a class of 50 pupils, 24 like oranges, 23 like apples and 7 like the two fruits.

1. How many do not like oranges and apples    
2. What percentage of the class like apples only

USING VENN DIAGRAM TO SOLVE PROBLEMS INVOLVING THREE SETS

Examples:

1. In a survey of 290 newspaper readers, 181 of them read the Daily Times, 142 read the Guardian, 117 read the Punch and each read at least one of the papers, If 75 read the Daily Times and the Guardian,60 read the Daily  Times and Punch and 54 read the Guardian and the Punch.

1. Draw a Venn diagram to illustrate the information
2. How many read:

1. all the three papers.
2. exactly two of the papers.
3. exactly one of the papers.
4. the Guardian only.

Solution

            n (E)= 290

n(P)= 117

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 – (60 – 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 = 290

251 + X = 290

X = 290 – 251

X= 39

b(i): number of people who read all the three papers = 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

= 62 +3X    = 62 + 3(39) 

= 62 + 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 follows:

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

Solution

1. 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 + 11 + 9 + 10 + 7 + 3 = 94

1. Number of students who like exactly two subject = 11 + 9 + 10 = 30

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

1. Draw a Venn diagram to illustrate the information.
2. Find the number of people that has both cars and bicycles.
3. How many people have either one of the three vehicles?

1.              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. In a senior secondary school, 80 students play hockey or football. The numbers that play football is 5 more than twice the number that play hockey. If 5 students play both games and every students in the school plays at least one of the games. Find:

1. The number of students that play football
2. The number of students that play football but not hockey
3. The number of students that play hockey but  not football

1. A, B and C are subsets of the universal set U such that

U={0,1,2,3,4………….12}

A={X: 0≤ x <7}        B= {4,6,8,10,12}        C= {1

1. Draw a venn diagram to illustrate the information
2. Find (i) BƲC (ii) AB∩C

READING ASSIGNMENT

NGM SSS1,page 106, exercise 8d, numbers 11-17. 

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 school, 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 speaks Hausa, 16 speaks Igbo, 21 speaks Yoruba and each of the students speaks at least one of the three languages. If 8 speaks Hausa and Igbo, 11 speaks Hausa and Yoruba,6speaks 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 visited a store in 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 certain class, 22 pupils take one or more of Chemistry, Economics and Government. 12 take Economics (E), 8 take Government (G) and 7 take Chemistry (C). nobody takes Economics and Chemistry and 4 pupils takeEconomics and Government

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

1. How many pupils take

1. Both Chemistry and Government?   
2. Government only?