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ET THEORY
The word set is used to denote a collection of well defined objects
∙ Set are denoted by capital letters e.g. A, B, C, D etc
∙ The statement ‘’ x is an element of A’’ or ‘’ x belong to A’’ is written as x ∈ A
If x is not an element of A, we write x 
A
AImportance sets of the number system
IR: a set of real numbers (+, -) all numbers
IR+: Is a set of positive real numbers
IR–: Is a set of negative real numbers
Z: a set of integers. (+, -) whole numbers
Z+: a set of positive integers
Z–: a set of negative integers
Q: a set of rational numbers (rational ½ = 0.33333 – rational numbers, number repeats and terminate
N: a set of natural number (positive numbers starting from 1, 2, 3…… counting numbers)
SPECIFICATION OF A SET
There are two ways of specifying a set;
1. List its members (roster method)
2. Describing its elements by mathematical notation or actual words (builder notation).
Examples
1. Let A = 
specified in roster form, specify this by set builder notation
specified in roster form, specify this by set builder notation Solution
A is a set of all prime numbers less than 15
2. Let B = 
specified by set builder, specify by roster form
specified by set builder, specify by roster form Solution
Since x2 = 9, x = 3, x = -3
B = 
The general form of set builder notation
A = 
OR
A = 
E.g. A = 
QUESTIONS
1. Let A = 
a) Is 10 ∈ A NO
b) Is 11 ∈ A NO
c) Is 13 ∈ A NO
d) List all elements of A
A = 
2. Use the roster method to specify the following sets
a) A = {x ∈ Z: x + 3 = 5}
x + 3 = 5; x = 5 – 3, x = 2
A = 
b) B = 
B = 
c) C = 
x = -0.5 and x = 0.5
C= 
3. Specify the following in roster form
a) A = {y ∈ Z: y= 3K where K∈Z+ and K ≤ 6}
Solution
K = 
Y = 
A =
b) B = 
y = 
B =
BASIC CONCEPTS OF SET.
1. The set that does not contain any element is called an empty set, donated by Φ or { }
2. Universal set is a set which contains all elements under consideration. It is denoted by µ.
3. Equality; two sets are equal if they have same elements
i.e. If A = 
and B =
and B =
4. Equivalent; two sets are equivalent if they have same number of elements
i. e A =
and B = 
∴A≡B
and B = 
∴A≡B5. Subsets; A is a subset of B if every member of A is also a member of B. It is denoted by A
B
B6. Improper subset; suppose A = and B =
AB
and B =
A
B7. Proper subset. Suppose A = 
and B = 
AB
and B = 
AB Note i) 
(an empty set is subset of any set)
(an empty set is subset of any set) ii) A A (a set is subset of its own set)
A (a set is subset of its own set)Number of subsets in a set
Let S =
How many subsets does it have?
The subsets are: { } 
→There are 8 subsets of S.
If A = and If B = 
and If B = 
Subset of A are : 
Subsets of B are : 
Subsets of B are : 
Number of subsets of A= 2 Number of subset of B = 4
If a set has n members, the number of subsets = 2n
THE POWER SET
Is a set which contains all subsets of the given sets
If A =, subsets are
, subsets are 
Power set of A is given by S = 
Given B = 
The power set of B is given by
S = 
OPERATION OF SETS
1. UNION
The union of two sets A and B is denoted by AUB
– AUB =
– Is a set which have elements of set A or set B without repetition.
Examples
→If A = and B = 
and B = 
AUB = 
→If A =
and B = 
and B = 
AUB = 
2. INTERSECTION
– Is a set which have both elements contained in set A and set B
A∩B = {x:x∈A and x∈B}
Examples
→If A = and B =
and B =
AB =
B =
→If A = and B =
and B = 
AB = 
B = 
Here A and B are disjoint sets.
3. COMPLEMENT
The complement of Set A denoted by A′ is the set of all elements which are in universal set but not in A.
E.g. A =
µ=
A′ =
4. RELATIVE COMPLEMENT
Relative complement of A with respect to set B is denoted by A’ B or A – B and is defined as follows
A
B = 
B = 
Example
A =
B = 
Then AB = 
B = 
BA = 
A = 
5. THE SYMMETRIC DIFFERENCE
All elements which are either in set A or set B but not both
– The symmetric difference of A and B is denoted by A 
B
BA B = 
B = 
Examples
A =
B =
A B =
B = 
QUESTIONS
1. List the subsets of the following sets
a) A =
b) B =
2. Let A = 
Write down the subsets of A
3. Which of the following are true and which are false?
a) Φ Φ b) 0 = Φ c) Φ∈ d) Φ ∈
Φ b) 0 = Φ c) Φ∈ 
d) Φ ∈ 4 . Let A = 
a) Is 
∈ A
∈ A b) Is 2 ∈ A
c) Is ∈ A
∈ A d) Is 
A
A e) Is 
f) Is 
5. Let µ be the set of all positive integers, A is the set of all even integers and B is a set of all odd integers. What are sets?
a) A B b) A B c) A B d) A’ e) B’ f) AB
B b) A 
B c) A B d) A’ e) B’ f) ABQUESTIONS
1. Let µ be the universal set and Φ be an empty set. What are
a) Φ = µ
b) µ = Φ
c) µ – Φ = µ
d) Φ – µ = Φ
e) µ ∩ Φ = Φ
f) µ Φ = µ
Φ = µ2. Let A be subset of the universal set µ. What are the following?
a) A Φ = A
Φ = A b) A A = A
A = A c) A Φ = Φ
Φ = Φ d) A A = A
A = A e) A µ = A
µ = A f) A µ = µ
µ = µg) A ∩ A’ = Φ or {}
h) A A‘= µ
A‘= µ i) A µ = A‘
µ = A‘ j) A Φ = A
Φ = A3. Let A and B be subsets of a universal set µ. Suppose A B. What are;
B. What are; a) A U B = B
b) A B = A
B = ASET INTERVAL ON THE NUMBER LINE
1. Let A = 
and B={x∈IR:-7< x ≤ 3}Represents these set intervals on two separate number lines
and B={x∈IR:-7< x ≤ 3}Represents these set intervals on two separate number lines Solutions
For A = 
For B = 
Examples
Using the sets A and B defined above, state and represents the following sets on same number line
a) A B b) A′ c) B′ d) A U B′
B b) A′ c) B′ d) A U B′ Solutions
a) A B
B
A B = 
B = 
b) A′
A′ = 
c)B′
c)B′
B′ = 
a)
(d)A U B′
(d)A U B′
A U B′ =
QUESTION
i) Represent the above sets on one number line
ii) Draw and state each of the following sets on separate number lines
a) A ∩ B b) A ∪B c) B′ d) A∩B′
Solution
(i)
Solution
(i)
(ii)(a) A
b) A U B
c) B′
QUESTIONS.
1. Represents and then draw on one number line the following set interval
Using the above set interval, represent and state the following
i) A B ii) A C iii) C B‘ iv) (AB) C
B ii) A C iii) C B‘ iv) (AB) CVENN DIAGRAMS
Sets can be represented in the form of diagrams called Venn diagrams
– The universal set is represented by a rectangle
– Subsets of U are represented by a circle in universal set
Uses of Venn diagram
i) To illustrate sets identity
ii) To find number of members in a given set
1. Illustration of set identity
Example Illustrate by use of Venn diagram (A U B) A = A
A = A Solution.
Two different methods can be used
i) Shading method
ii) Numbering of disjoint subsets
i) Shading method, i.e. to show (A B) ∩ A = A
B) ∩ A = A L. H. S → (A B) ∩ A
B) ∩ A Shade (A B) by vertical lines
B) by vertical lines Shade (A B) A by horizontal lines
B) A by horizontal lines Now (A B) A = region shaded
B) A = region shaded = A
= R. H. S
∴ (A
B) A = A ii) Numbering of disjoint
Solutions
L. H. S = (A B) A
B) A Now A B = subsets 1, 2, 3
B = subsets 1, 2, 3 But A = sub 1, 2
(A B) A = subsets 1, 2
B) A = subsets 1, 2 =A
= R. H. S
Example
Use Venn diagram to show A (B C) = (A B) (A C)
(B C) = (A B) (A C)Solution
L. H. S = A U (B C)
C)Now B C subsets 5, 6
C 
subsets 5, 6A U (B C)
Subsets 1, 2, 5, 4 and 6
C)
Subsets 1, 2, 5, 4 and 6R. H. S = (A U B) (A U C)
(A U C)A U B
subsets 1, 2, 3, 4, 5, 6
subsets 1, 2, 3, 4, 5, 6A U C subsets 1, 2, 3, 4, 5, 6, 7
subsets 1, 2, 3, 4, 5, 6, 7(A U B) ∩ (A U C) = 1, 2, 5, 4, 6
A (B C) = (AB) (A C)
(B C) = (AB) (A C)QUESTION
Use a Venn diagram to show the following
i) (A B) A = A
B) A = Aii) A (B C) = (A B) (A C)
(B C) = (A B) (A C)LAWS OF ALGEBRA OF SETS
Set operations obey the following laws
1. Commutative laws
A U B = B U A
A B = B A
B = B A2. Associative laws
a) (A U B) U C = A U (B U C)
b) (A B) C = A (B C)
B) C = A (B C)3. Distributive laws
a) A U (B C) = (A U B) (A U C)
C) = (A U B) (A U C) b) A (B U C) = (A B) (A C)
(B U C) = (A B) (A C)4. De -Morgan’s laws
a) (A U B)′ = A′ B′
B′ b) (A B)′ = A′U B′
B)′ = A′U B′5. Identity laws
a) A µ = µ
µ = µ b) A µ = A
µ = A c) A Φ = A
Φ = A d) A Φ =Φ
Φ =Φ e) AΦ = A
f) AA = Φ
Examples
Use laws of algebra of set to simplify
1. (A (A B)′)′
(A B)′)′ Solution
(A (A B)′)′ ≡(A (A′ B′))′ De-Morgan’s law
(A B)′)′ ≡(A (A′ B′))′ De-Morgan’s law≡((A A′) B′ )′Associative law
A′) B′ )′Associative law≡ (Φ B′) Complement law
B′) Complement law≡ (Φ)′Identity law
≡ µ complement law
(A(A U B)′)′ = µExamples
Use the laws of algebra of sets to prove
(A (B C′)) C = (A C) (B C)
(B C′)) C = (A C) (B C) Solution
L.H.S (A (B C′)) C
(B C′)) C = (((A B) C′) C…….. Associative law
B) C′) C…….. Associative law =((A B) U C) (C′ C) ………distributive law
B) U C) (C′ C) ………distributive law = ((A B) C) (µ) …………complement law
B) C) (µ) …………complement law = (A B) C……………. identity law
B) C……………. identity law = (A C) (B C) ……………distributive law
C) (B C) ……………distributive law = R. H. S
Exercise
1. Use laws of algebra of set to simply
i) (A B) (A B’)
B) (A B’) ii) (A’ B’) (A B)
B’) (A B) iii) (A B) U (A – B)
B) U (A – B) iv) A (A B)
(A B)2. Use laws of algebra to prove
i) (Z W)′ W = Φ
W)′ W = Φ ii) (X Y’) (X Y) (Y X′) = X Y
Y’) (X Y) (Y X′) = X Y iii) (A – B) A = A
A = A Note
A – B = A B′ by definition
A – B = A
B′ by definitionNumber of elements in a set
The number of elements in set A is denoted by n (A)
Example
Let A be a set of all positive odd integers which are less than 10. Find n (A)
Solution
A = {1, 3, 5, 7, 9}
Now n (A) = 5
Examples
Let A ={x ∈ IR:x2-x-2=0}. Find n (A)
Solution
Note
i) The number of elements of a set is defined only for a finite set
ii) If A U then the number of
elements of A′ is n(A′) = n(µ) – n(A)
U then the number ofelements of A′ is n(A′) = n(µ) – n(A)
Example
If A U and B U then show that n (A B) = n(A) + n(B) – n(A B)
U and B U then show that n (A B) = n(A) + n(B) – n(A B) Proof
Refer to the Venn diagram below
Represents the number of elements in disjoint subset as follows
Let n (A B′) = a n (A′ B) = c
n (A B) = b
Let n (A
B′) = a n (A′ B) = c n (A
B) = b R. H. S = n (A) + n (B) – n (A B)
B) = (a + b) + (b + c) – b
= a + 2b + c – b
= a + b + c
n (A B)
B) L. H. S
EXAMPLE
1. Given n (X) = 18, n (Y) = 26, n (X ∩ Y) = 12. Find n (X Y)
Y)2. Given n (S T) = 19, n (s) = 15. n (S T′) = 10. Find n(S T)
T) = 19, n (s) = 15. n (S T′) = 10. Find n(S T)3. Given n (A B) = 15
B) = 15 n (A B) = 16
B) = 16 n ((A B)′) = 4
B)′) = 4 n (A – B) = 8
Find i) n (A) ii) n (A B′) iii) n (µ) iv) n (A′ B)
B′) iii) n (µ) iv) n (A′ B)Solutions
1. n (X Y) = n (X) + n (Y) – n (X Y)
Y) = n (X) + n (Y) – n (X Y)=18 + 26 – 12
= 32
n (X Y) = 32
Y) = 322. n(S T) = 19, n(S) = 15, n(S T’) = 10
T) = 19, n(S) = 15, n(S T’) = 10 i) n(S T) =?
T) =?n(S T) = n(S) – n(S T’)
T) = n(S) – n(S T’) = 15 – 10
= 5
n( S T) = 5
T) = 53. n(A B) = 5, n(A B) = 16, n(A B)′ = 4, n(A – B) = 8
B) = 5, n(A B) = 16, n(A B)′ = 4, n(A – B) = 8i) n(A) =? ii) n (A B′) iii) n(µ) iv) n (A′B)
B′) iii) n(µ) iv) n (A′B)Solutions
n (A) = n(A – B) + n(A B)
B) = 8 + 5
= 13
ii) n (A B’) = n (A) + n(B′)
B’) = n (A) + n(B′) = 13 + 4
= 17
n(A B’) = 17
B’) = 17 iii) n(µ) = n(A B) + n(( A B))′
B) + n(( A B))′ = 16 + 4
= 20
n(µ) = 20
iv) n(A′ B) = n(B) – n(A B)
B) = n(B) – n(A B) n(A′ B) = n(A B) – n(A) + n(A B) – n (n (A B))
B) = n(A B) – n(A) + n(A B) – n (n (A B)) n(A′ B) = 16 – 13
B) = 16 – 13 n (A′ B) = 3
B) = 34. By using n (A B) = n(A) + n(B) – n(A B) show that;
B) = n(A) + n(B) – n(A B) show that;n(A B C) = n(A) + n(B) + n(C) – n(A B) – n(A C) – n(B C) + n(A B C)
B C) = n(A) + n(B) + n(C) – n(A B) – n(A C) – n(B C) + n(A B C)Solutions
Let B C = K
C = KL.H.S n(A B C) = n(A K)
B C) = n(A K) = n(A) + n(K) – n(A K)
K) = n(A) + n(B C) – n(A (B C))
C) – n(A (B C)) = n(A) + n(B) + n(C) – n(B C) – n((A B) (A C))
C) – n((A B) (A C))= n(A) + n(B) + n(C) – n(B C) – (n(A B) + n( A C) – n((A B)(A C))
C) – (n(A B) + n( A C) – n((A B)(A C)) = n(A) + n(B) + n(C) – n(B C) – n(A B) – n(A C) + n(A B C)
C) – n(A B) – n(A C) + n(A B C)Questions
There are 26 animals in zoo, 5 animals eat all type of food in the zoo i.e. grass, meat and bones. 6 animals eat grass and meat only, 2 animals eat grass and bones only, 4 animals eat meat and bones only. The number of animals eating one type of food only is divided equally between the three types of food.
i) Illustrate the above information by a labeled Venn diagram
ii) Find the number of animals eating grass
Solutions
Let M set of animals that eat meat
set of animals that eat meat Let B set of animals that eat bones
set of animals that eat bones Let G set of animals that eat grass
set of animals that eat grass 3
+ 6 + 5 +
4 + 2 = 26
+ 6 + 5 +4 + 2 = 26
3 + 17 = 26
+ 17 = 26
= 3 ii) Number of animals eating grass
= 6 + 5 + 2 + 3
= 16 animals
Questions
1. A class has 15 boys and 15 girls. In the class 20 students are studying science, 14 students are studying math, 10 boys are studying science, 10 boys are studying math, 8 boys are studying both math and science, 4 girls are studying neither math nor science.
1. A class has 15 boys and 15 girls. In the class 20 students are studying science, 14 students are studying math, 10 boys are studying science, 10 boys are studying math, 8 boys are studying both math and science, 4 girls are studying neither math nor science.
Find i) How many students study math only?
ii) How many students study science only?
iii) How many students study both math and science?
ii) How many students study science only?
iii) How many students study both math and science?
2. In a class of 35 students each students each student takes either one of two subjects (physics, chemistry and biology). If 13 students take chemistry, 22 students take physics,17 students take biology, 6 students take both physics and chemistry and 3 students take both biology and chemistry. Find the number of students who take both biology and physics.
Solutions
Since there are 15 girls
10 – + + 4 – + 4 = 15
+ + 4 – + 4 = 15 18 – = 15
= 15 = 3 i) Students who study math only = 2 + 1
= 3 students
ii) Students who study science only = 2 + 7
= 9 students
iii) Students who study both math and science = 8 + 3
= 11 students
QUESTIONS
1. In a certain college apart from other discipline, no students is allowed to study less than two of the subjects, finance, accounting and economics, 150 students study finance, 110 study accounting, 80 study economics and 20 study three subjects
i) How many students study two of the named subjects?
ii) How many study finance or accounting or economic?
ii) How many study finance or accounting or economic?
2. One poultry farm in Dar produces three types of chicks and in six months report revealed that out of 126 of its regular customers, 65 bought broilers, 80 bought layers and 75 bought cocks, 45 bought layers and cocks, 35 bought broilers and cocks, 10 bought broilers only, 15 bought layers only and bought cocks only, 6 of the customers did not show up.
i) How many customers bought all the three products?
ii) How many customers bought exactly two of the products?
ii) How many customers bought exactly two of the products?
3. An investigator was paid sh. 100 per person interviewed about their likes and dislikes on a drink for lunch. He reported 252 responded positively coffee, 210 liked tea, 300 liked soda, 80 liked tea and soda, 60 liked coffee and soda. 50 liked all three, while 120 people said they not like any drink at all. How much should the investigator be paid coffee and tea 60 people?
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