MATHEMATICS O LEVEL(FORM TWO) NOTES

SETS

A set is a group/ collection of things such as a herd of cattle, a pile of books, a collection of trees, a shampoos bees and a fleck of sheep

Description of sets

-A set is described/denoted by Carl brackets { } and named by Capital letters

Examples

If A is a set of books in the library then A is written as

A= {All books in the library} and read as A is a set of all books written in the library

-The things/objects In the set are called Elements or members of the set

Example

1. If John is a student of class B, then John is a member of class B and shortly denoted as £B.

2. If A= {1,2,3} then 1

A, 2

A and 3

The number of elements in a set is denoted by n(A)

Example

1. If A= {a, e, i, o, u} then n (A) =5

Example

If A is a set of even, describe this set by

a) Words

b) Listing

c) Formula

Solution:

a) By words;

A= {even numbers}

b) By Listing

A= {2, 4, 6, 8…}

c) By Formular

A= {x: x= 2n} where n= {1, 2, 3…} and is read as A is a set of all element x such that x is an even number.

2. Describe the following sets by Listing

A= {whole numbers between 1 and 8}

Solution:

A= {2, 3, 4, 5, 6, 7}

3. Write the following sets in words

A= {an integer < 10}

Solution:

A= {integers less than ten} or A is a set of integers less than ten

TYPES OF SETS

Finite set; Is a set where all elements can be counted exhaustively.

Infinite set: An Infinite set is a set that all of its elements cannot be exhaustively counted

Example

B= {2, 4, 6, 8…}

An Empty set Is a set with no elements. An Empty set is denoted by { } or Ø

Example

If A is an Empty set then can be denoted as A= { } or A= Ø

Exercise

1. List the elements of the named sets

A= {x: x is an odd number < 10}

A= {1, 3, 5, 7, 9}

B= {days of the week which began with letter S}

B= {Saturday, Sunday}

C= {Prime numbers less than 13}

C= [2, 3, 5, 7, 11}

2. Write the named sets in words

B= {x: x is an odd number < 12}

B is a set of x such that x is an odd number less than twelve

E= {x: x is a student in your class}

E is a set of x such that x is a student in your class

3. Write the named sets using the formula methods

A= {all men in Tanzania}

A= {x: x is all men in Tanzania}

B= {all teachers in your school}

B= {x: x is all teachers in your school}

C= {all regional capital in Tanzania}

C= {x: x is all regional capital in Tanzania}

D= {b, c, d, f, g…}

D= [x: x is a consonant}

COMPARISON OF SETS

-SET may be equivalent, equal or one to be a subset of other

-Equivalent sets are sets whose members (numbers) match exactly

Example

A= {2, 4, 6, 8} and B= {a, b, c, d}

Then A and B are equivalent

The two sets can be matched as

C:thlbcrtz__i__images__i__d9.png

Generally if n(A) = n(B) then A and B are equivalent sets

Example

If A= {1, 2, 3, 4} and B= {1, 2, 3, 4} since n(A) = n(B) and the elements are alike then set A is equal to set B

Subset: Given two sets A and B, B is said to be a subset of A. If all elements of B belongs to A

Example

If A= {a, b, c, d, e} and B= {a, b, c, e}, Set B is a subset of A since all elements of set B belongs to set A. But set B has less elements than set A. Then set B is a proper subset of set A

and A is a super set of B.

If A= B then either A is an improper subset of B or B is an improper subset of A.

Symbolically written as A⊆B or B⊆A

Note: an empty set is a subset of any set

-The number of subset in a set is found by the formula 2n where n= number of elements of a set

Example

1. List all subset of A= {a, b}

Solution:

2n , n = number of element in a set.
So 22 = 4

The number of subset = 4

The subset of A are { }, {a}, {b}, {a, b}

2. How many subset are there in A= {1, 2, 3, 4}

Solution:

The number of subset= 24= 16

UNIVERSAL SET [U]

-Is a single sets which contains all elements sets under consideration for example the set of integers contains all the elements of sets such as odd numbers, even numbers, counting numbers, and whole numbers. In this case the set of integers is the Universal set.

Exercise

1. Which of the following sets are

a) Finite set

b) Infinite set

c) Empty set

A= {Nairobi, Dar es Salaam}

B= {2, 4, 6…36}

E= {All mango trees in the world}

F= {x: x is all students aged 100 years in your school}

H= {1, 3, 5, 7}

D= {all lions in your school}

I= Ø

Solution:

a) Finite set are

A= {Nairobi, Dar es Salaam}

B= {2, 4, 6…36}

H= {1, 3, 5, 7}

Infinite set

E= {All mango trees in the world}

F= {x: x is all students aged 100 years in your school}

b) Empty sets are

D= {all lions in your school}

I= Ø

2. In each of the following pairs of sets shown by matching whether the pairs are equivalent or not Equivalent are:

A= {a, b, c, d} and B= {b, c, d, e}

Which are not equivalent are:

B= {Rufiji, Ruaha, Malagarasi} and C= {lion, leopard}

B and C are not equivalent.

3. Which of the following sets are equal

A= {a, b, c, d}, B= {d, a, b, c}, C= {a, e, I, o, u}, D= {a, b, c, d}, E= {d, c, b, a}, F= {a, e, b, c, d}

Solution

A, B, D and E are equal

4. List all subsets of each of the following sets

a) A= 1

The number of subset = 21= 2

Therefore; The subset of A are { }, {1}

b) B= Ø

Therefore; number of subsets is { }

c) C = {Tito, Juma}

Number of subset = 22=4

Therefore; the subsets of C are { }, {Tito}, {Juma}, {Tito, Juma}

5. Name the subsets of each pair by using the symbol ⊂

a) A= {a, b, c, d, e, f, g, h} and B= {d, e, f}

Therefore = B⊂A

b) A= {2, 4} and D= {2, 4, 5} = A⊂D

c) A= {1, 2, 3, 4 …} and B= {2, 4, 6, 8…} = A⊆B

6. Given G = {cities, towns and regions of Tanzania} which of the following sets are the subsets of G?

A= {Nairobi, Dar es Salaam}

B= {Dodoma, Mombasa, Mwanza}

C= { }

D= {Arusha, Iringa, Bagamoyo}

E= {Mbeya, Tunduru, Ruvuma}

Therefore; the subsets of G are C, D, E

7. Which of the following sets are the subsets of K given that K= {p, q, r, s, t, u, v, w}

A= {p, s, t, x}

B= {q, r, d, t}

C= { }

D= {p, q, r, s, t, u, v, w}

E= {a, b, c, d}

F= {s, v, q}

Therefore; the subsets of K is D, C, F

8. What is n(A) if A= { }

n(A) = 0

9. Write in words the universal set of the following sets

a) A= {a, b, c, d}

The universal set of A is a set of alphabets

b) B= {1, 2, 3, 4}

The universal set of set B is the set of natural numbers

OPERATION WITH SETS

UNION

The union of two sets A and B is the one which is formed when the members of two sets are putted together without a repetition. Thus the union is

, this union of A and B can be denoted as A

B is defined as x; X

A or X

Example

1. If A= {2, 4, 6} and B= {2, 3, 5} then A

B= {2, 4, 6}

{2, 3, 5}= {2, 3, 4,5, 6}

2. Find A

B when A= {a, b, c, d, e, f} and B= {a, e, I, o, u}

Solution:

B= {a, b, c, d, e, f, I, o, u}

INTERSECTION

The Intersection of two sets A and B is a new set formed by taking common elements. The symbol for intersection is “

”

Example

1. A= {1, 2, 3, 4, 5}, B= {1, 3, 5} then A

B= {1,3,5}

2. Find A

B if A= {a, e, i, o, u}, B= {a, b, c, d, e, f} then A

B= {a, e}

COMPLEMENT OF A SET

If A is a subset of a universal set, then the members of the universal set which are not in A, form compliment of A denoted by AÎ„

Example

= {a, b, c … z} and A= {a, b} then AÎ„= {c, d, e, … z}

Given that U= {15, 45, 135, 275} and A= {15} find AÎ„
Solution:

A’= {45, 135, 275}

JOINT AND DISJOINT SETS

JOINT SETS Are sets with common elements

E.g. A= {1, 2, 3, 5}, D= {1, 2} then A and D are joint sets since {1, 2} are common elements

DIS JOINT SETS Are sets with no elements in common

For example A= {a, b, c} and B= {1, 2, 3, 4} then A and B are disjoint sets since they do not have a common element

EXERCISE

1.Find

a) Union

b) Intersection of the named sets

1. A= {5, 10, 15}, B= {15, 20}

a) A

B = {5, 10, 15, 20}

b) A

B= {15}

2. A= { }, B= {14, 16}

a) A

B= { , 14, 16}

b) A

B= { }

3. A= {First five letters of the English alphabet}, B= {a, b, c, d, e}

a) A

B= {a, b, c, d, e}

b) A

B= {a, b, c, d, e}

4. A= {counting numbers}, B= {prime numbers}

a) A

B= {counting numbers}

b) A

B= {prime numbers}

5. A = {o, }, B= { }

VENN DIAGRAM

-Are the diagrams (ovals) devised by John Venn for representation of sets

Example

If A= {a, b, c} can be represented as

µ is the uni
versal set, in this case is the set of all English alphabets. If the set have any elements in common, the ovals over lap for example, If A= {a, b, c} and B= {a, b, c, d} then it can be represented as

Disjoint sets also can be represented on a Venn diagram

Example: If A= {a, b}, B= {1, 2} the relation A and B is as follows

Examples

If A is a subset of B, represent the two sets on a Venn diagram

C:thlbcrtz__i__images__i__img645.jpg

Represent A= {2, 3, 5}, B= {2, 5, 7} C= {2, 3, 7} in a Venn diagram

Solution:

Represent AUB in a Venn diagram given that A= {1, 2}, B= {1, 3, 5}

Solution:

If set A and B have same elements in common, represent the following in a Venn diagram

a) A

b) A

Solution:

In a certain primary school 50 pupils were selected to form three schools teams of football, volleyball and basketball as follows

30 pupils formed a football team

20 pupils formed a volleyball team

25 pupils formed a basketball team

14 play both volleyball and basketball

18 pupils play football and basketball

8 pupils play all of the three games

7 pupils play football only

Represent this information in a Venn diagram

Solution:

A and B are sets such that n(A

B)=4 and n(A

B)=6 if A has 4 elements

a) How many elements are there in B?

b) Which set is the subset of the other

Solution:

(a). 6 elements are in B

(c). A⊂B

In general the number of elements in two sets is connected by the formula

n(A

B)= n(A) + n(B) – n(A

Exercise:

1. Represent the following in Venn diagrams

a) A={a, b, c, d}

b) A⊂B

c) A= {a, b, c} and B= {a, b, c}

d) A= {1, 2, 3} and B= {4, 6, 8}

2. Write in words the relationship between the two sets shown in the figure below

-Their relationship is A⊂B

3. Describe in set notation the meaning of the shaded regions in the following Venn diagrams

A= {a, b, c}

b) A

B= {a, b, c}

4. In a boys school of 200 students, 90 play football, 70 play basketball, and 30 play Tennis. 26 play basketball and football, 20 play basketball and Tennis, 16 play football and Tennis, while 10 play all three games. How many students in school play none of the three games

4+10+34+6+10+16+58+N= 200

138+N= 200

N=200-138

N=62

62 students play none of the games

COMPLEMENT OF A SET

If A is a subset of a universal set, then the compliment of set A may be represented in a Venn diagram

Example

1. Show in a Venn diagram that (AUB)’

Solution:

B)Î„

B)’ = members of outside A

B’

3. Represent AÎ,

, µ in a Venn diagram and shade the required region

A is a subset of Universal set

WORD PROBLEMS

Examples

1. In a certain school of 120 students, 40 learn English, 60 learn Kiswahili and 30 learn both Kiswahili and English. How many students learn

a) English only

b) Neither English nor Kiswahili

Solution:

Let µ = {students in a school}

A= {Students learning English}

B= {Students learning Kiswahili}

a) n(A) – n(A

B) = number of students learning English only

40 – 30 = 10

Therefore the number of students learning English only is 10

b) =n(µ)-[n(A)+ n(B)- n(A

B)]

= 120-[40+60-30]

=120-70

=50

50 students learn neither English nor Kiswahili

Alternatively

By Venn diagram

C:thlbcrtz__i__images__i__img2422.jpg

a) 10 students learn English only

2. In a certain school 50 students eat meat, 60 eat fish and 25 eat both meat and fish. Assuming that every students eat meat or fish, find the total members of students in a school

Solution:

Let µ= {total number of students}

A= {students eating fish}

B= {students eating meat}

n(A

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

n(A

B)=50+60-25

n(A

B)= 85 students

There are 85 students in a school

Alternatively

By Venn diagram

85 students were in school

3. There are 24 men at a meeting, 12 are farmers, 18 are soldiers, 8 are both farmers and soldiers

a) How many are farmers or soldiers

b) How many are neither farmers nor soldiers

Solution:

NB; both/and means “intersection”

By Venn diagrams

a) 22 men are soldiers or farmers

b) 2 are neither farmers nor soldiers

4. In an examination, 120 candidates offered math, 94 English and 48 offered both math and English. How many candidates offered English but not math assuming that every candidate offered one of the subjects or both math and English

Solution:

By Venn diagram

46 students offered English but not math

EXERCISE

1. A class shows that 15 of the students play basketball, 11 play netball and 6 play both basketball and netball. How many students are there in a class? If every student plays at least one game

Solution:

n(A

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

n(A

B)=15+11-6

n(A

B)=20

There are 20 students in the class

2. In a class of 20 pupils, 12 pupils study English but not History, 4 study History but not English and 1 who study neither English nor History. How many study History

Solution:

Let A= {pupils who study English}

B= {pupils who study History}

12 + x + 4 = 20

X = 3

History = x + 4 =7

7 pupils study History

3. At a certain meeting 30 people drank Pepsi, 60 drank Coca-Cola, and 25 drank both Pepsi and Coca-Cola. How many people were at the meeting assuming that each person took Pepsi or Coca-Cola

Solution:

Let A= {drank Pepsi}

B= {drank Coca-Cola}

n(A

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

=30+60-25

=65

65 people were at the meeting

4. Represent (A

C) on a Venn diagram

C:thlbcrtz__i__images__i__img3014.jpg

Represent (A

6. If set A and B have the same common elements represent

a) A

b) A

B in a Venn diagram

In a school of 160 pupils, 50 have bread for breakfast and 80 have sweet potatoes. How many pupils have neither Bread nor potatoes assuming that none take bread and sweet potatoes

30 pupils have neither Bread nor Potatoes

8. Every Man in a certain club owns a Land Rover or a car. 23 men own Land Rover, 14 own cars and 5 owns both Land Rovers and cars. How many men are in a club?