PU
N
LN
Chapter 1
SE
SETS, RELATIONS AND SEQUENCE
RI
ES
Chapter Overview
CO
We start this chapter with set as we did in Ordinary General Level Mathematics. We will then
introduce special sets namely ordered power sets, ordered pairs and Cartesian products. Then
a discussion is given to binary relations or relations.The chapter will be concluded with set
PY
of numbers, sequence and series in which again emphasis will be on arithmetic and geometric
progressions and arithmetic and geometric series. Hence by the end of this you should be able
RI
to do the following:
G
1. describe a set
H
2. perform algebra of set
T
3. draw venn diagrams
RE
4. define a relation
SE
5. find domain, range, source and target sets of a relation
6. describe set of numbers
R VE
7. define a sequence and perform relevant calculations with respect to sequence
8. Use Principle of Mathematical Induction to do some prrofs
D
1.1 Elementary Set Theory
20
25
Definition 1.1 A set is a well defined collection of objects.
1
,PU
The objects which make up the set are called members or elements of the set. Generally
we denote sets with capital(upper case) letters A, B, . . ., Z and the elements with small(lower
N
case) letters a,b, . . .,z. Note that the elements need not always be a letter.
LN
1.1.1 Membership and Description of a Set
We described sets in two ways namely:
SE
(i)Listing or Roaster Method:In this method, if possible one directly names each element
of the set separated by commas enclosed between braces. Examples are V = {a, e, i, o, u},
RI
E = {2, 4, 6, . . . , 100}. That is set of English Vowels and positive even integers lees than or
equal to 100 respectively.
ES
(ii) Set Builder or Specifier Notation: Here the quality or property of the elements are
given in between braces. Thus we have
V = {x : x is an Englis Vowel} and E = {y/y is an even integer ≤ 100}.
CO
1.1.2 Equality of Sets
PY
A set S is said to be equal to another set T if they have same elements. I.e. for all objects
x ∈ S, then x ∈ T and vice-versa or x ∈ S if and only if(iff or ⇔)x ∈ T . Otherwise S ̸= T .
RI
Hence
{a, b, 2, 3} = {2, a, 3, b}- order of elements immaterial or does violate equality of sets.
G
{c, 2, c} = {2, c, 2, c, c, 2, 2, c} - repetition of elements immaterial or does not violate equality of
H
sets
T
{a, b} = {2, b}.
RE
1.1.3 Subsets
SE
If every elements of a set S is an element of a set T , then S is called a subset of T (and T is a
superset of S) and this is denoted by S ⊂ T (S is contained or smaller or included in T ). And
T ⊃ S (T contains or is bigger than or includes S)
R
For example,
VE
DEDE or DEED ⊂ LED
ADD ⊂ DAD
D
CAT ⊈ HAT
20
Proper Subset: If Set A is a subset of Set B, but Set A is not equal to Set B, then A is said
25
to be a proper subset of B, denoted as A ⊂ B.
2
,PU
Improper Subset: If Set A is a subset of Set B, and Set A is equal to Set B, then A is said
to be an improper subset of B. This is denoted as A ⊆ B.
N
Trivial Subset:Is a set who is always a subset of any given set X. The empty set and the
given set X is the only trivial subsets of any given set X.
LN
Remark 1.2
(i) Two seat A and B are equal if and only if (⇔) they are subsets of each other. I.e. A = B ⇔
SE
A ⊂ B and B ⊂ A (ii) the inclusion property satisfies:
(◦)A ⊂ A( reflexive property or reflexivity)
RI
(◦◦)A ⊂ B, B ⊂ C, then A ⊂ C( transitive property or transitivity)
ES
1.1.4 Algebra of Sets
(a) Empty Set: Is a set which has no element. It is usually denoted by ϕ or {}. For example,
CO
the set of female Vice-Chancellors of the University of Nigeria from 1960 to 2024.
(b). Union of Sets; The union of two or more sets is a set comprising all the elements which
appears in at least of the sets whose union is being found. Hence if A and B are two sets,
PY
then the union of A and B is given by
RI
A ∪ B = {x : x ∈ A or x ∈ B} (1.1)
If χ is a any non empty collection of sets, the union of all sets in χ, is the set of all elements
G
S S
which are elements of at least a set in χ and is denoted by {A : A ∈ χ} or A. Hence
H
A∈χ
T
[ [
χ= {A : A ∈ χ} = {x : x ∈ A for some A ∈ χ} (1.2)
RE
(c). Intersection of Sets; The intersection of two or more sets is a set comprising all the elements
common to the sets. Hence if A and B are two sets, then the union of A and B is given by
SE
A ∩ B = {x : x ∈ A andr x ∈ B} (1.3)
For any non empty collection F of sets , the set of all elements which are elements of every set
R
T T
of F is the intersection of all sets in χ, and is denoted by {A : A ∈ F} or A. Therefore
VE
A∈F
\ \
F= {A : A ∈ F} = {x : x ∈ A ∀ A ∈ F} (1.4)
D
(d) Complement of a Set: If A and B are sets, then A’s complement with respect to B or A
20
difference B or A minus B denoted by A\B or A − B is defined as
25
A − B = {x ∈ A : x ∈
/ B} (1.5)
3
N
LN
Chapter 1
SE
SETS, RELATIONS AND SEQUENCE
RI
ES
Chapter Overview
CO
We start this chapter with set as we did in Ordinary General Level Mathematics. We will then
introduce special sets namely ordered power sets, ordered pairs and Cartesian products. Then
a discussion is given to binary relations or relations.The chapter will be concluded with set
PY
of numbers, sequence and series in which again emphasis will be on arithmetic and geometric
progressions and arithmetic and geometric series. Hence by the end of this you should be able
RI
to do the following:
G
1. describe a set
H
2. perform algebra of set
T
3. draw venn diagrams
RE
4. define a relation
SE
5. find domain, range, source and target sets of a relation
6. describe set of numbers
R VE
7. define a sequence and perform relevant calculations with respect to sequence
8. Use Principle of Mathematical Induction to do some prrofs
D
1.1 Elementary Set Theory
20
25
Definition 1.1 A set is a well defined collection of objects.
1
,PU
The objects which make up the set are called members or elements of the set. Generally
we denote sets with capital(upper case) letters A, B, . . ., Z and the elements with small(lower
N
case) letters a,b, . . .,z. Note that the elements need not always be a letter.
LN
1.1.1 Membership and Description of a Set
We described sets in two ways namely:
SE
(i)Listing or Roaster Method:In this method, if possible one directly names each element
of the set separated by commas enclosed between braces. Examples are V = {a, e, i, o, u},
RI
E = {2, 4, 6, . . . , 100}. That is set of English Vowels and positive even integers lees than or
equal to 100 respectively.
ES
(ii) Set Builder or Specifier Notation: Here the quality or property of the elements are
given in between braces. Thus we have
V = {x : x is an Englis Vowel} and E = {y/y is an even integer ≤ 100}.
CO
1.1.2 Equality of Sets
PY
A set S is said to be equal to another set T if they have same elements. I.e. for all objects
x ∈ S, then x ∈ T and vice-versa or x ∈ S if and only if(iff or ⇔)x ∈ T . Otherwise S ̸= T .
RI
Hence
{a, b, 2, 3} = {2, a, 3, b}- order of elements immaterial or does violate equality of sets.
G
{c, 2, c} = {2, c, 2, c, c, 2, 2, c} - repetition of elements immaterial or does not violate equality of
H
sets
T
{a, b} = {2, b}.
RE
1.1.3 Subsets
SE
If every elements of a set S is an element of a set T , then S is called a subset of T (and T is a
superset of S) and this is denoted by S ⊂ T (S is contained or smaller or included in T ). And
T ⊃ S (T contains or is bigger than or includes S)
R
For example,
VE
DEDE or DEED ⊂ LED
ADD ⊂ DAD
D
CAT ⊈ HAT
20
Proper Subset: If Set A is a subset of Set B, but Set A is not equal to Set B, then A is said
25
to be a proper subset of B, denoted as A ⊂ B.
2
,PU
Improper Subset: If Set A is a subset of Set B, and Set A is equal to Set B, then A is said
to be an improper subset of B. This is denoted as A ⊆ B.
N
Trivial Subset:Is a set who is always a subset of any given set X. The empty set and the
given set X is the only trivial subsets of any given set X.
LN
Remark 1.2
(i) Two seat A and B are equal if and only if (⇔) they are subsets of each other. I.e. A = B ⇔
SE
A ⊂ B and B ⊂ A (ii) the inclusion property satisfies:
(◦)A ⊂ A( reflexive property or reflexivity)
RI
(◦◦)A ⊂ B, B ⊂ C, then A ⊂ C( transitive property or transitivity)
ES
1.1.4 Algebra of Sets
(a) Empty Set: Is a set which has no element. It is usually denoted by ϕ or {}. For example,
CO
the set of female Vice-Chancellors of the University of Nigeria from 1960 to 2024.
(b). Union of Sets; The union of two or more sets is a set comprising all the elements which
appears in at least of the sets whose union is being found. Hence if A and B are two sets,
PY
then the union of A and B is given by
RI
A ∪ B = {x : x ∈ A or x ∈ B} (1.1)
If χ is a any non empty collection of sets, the union of all sets in χ, is the set of all elements
G
S S
which are elements of at least a set in χ and is denoted by {A : A ∈ χ} or A. Hence
H
A∈χ
T
[ [
χ= {A : A ∈ χ} = {x : x ∈ A for some A ∈ χ} (1.2)
RE
(c). Intersection of Sets; The intersection of two or more sets is a set comprising all the elements
common to the sets. Hence if A and B are two sets, then the union of A and B is given by
SE
A ∩ B = {x : x ∈ A andr x ∈ B} (1.3)
For any non empty collection F of sets , the set of all elements which are elements of every set
R
T T
of F is the intersection of all sets in χ, and is denoted by {A : A ∈ F} or A. Therefore
VE
A∈F
\ \
F= {A : A ∈ F} = {x : x ∈ A ∀ A ∈ F} (1.4)
D
(d) Complement of a Set: If A and B are sets, then A’s complement with respect to B or A
20
difference B or A minus B denoted by A\B or A − B is defined as
25
A − B = {x ∈ A : x ∈
/ B} (1.5)
3
