Lecture notes Introduction to Abstract Algebra (MAT215)

SOUTHWESTERN UNIVERSITY, NIGERIA
MAT 215 – INTRODUCTION TO ABSTRACT ALGEBRA
Lecture Note by DANIEL Deborah O.

COURSE OUTLINE

 Basic Review of Set Theory
 Binary Relations
 Mapping
 Binary Operation
 Group Theory
 Rings
 Integral Domain
 Fields.

,BASCI REVIEW OF SET THEORY

Basic Review of set theory

Set: A set is any list or collection of well-defined object viewed as a single entity. Each
member of a set is called an element of the set. Conventionally, sets are denoted by
capital alphabetic letters 𝐴, 𝐵, 𝐶, 𝐷, … ete white small a lphabetic letters 𝑎, 𝑏, 𝑐, 𝑑, … etc
are used to denote the elements of a set Aset is usvally specified by listing all its elements
or by stating properties which characterize its elements in braces. If 𝐴 is a non empty set
and 𝑎 is an element of the set 𝐴, we write 𝑎 ∈ 𝐴. If an element 𝑏 is nota menber of the set
𝐵, we write b ∉ 𝐴

Example

(i) 𝐴 = {𝑥: 𝑥 2 + 8𝑥 + 15 = 0, 𝑥 ∈ 𝑍} where ℤ is the set of integers.
(ii) 𝐵 = {𝑥: 𝑥 ∈ ℕ and 𝑥 is a muttiple of 5}.i.e 𝐵 = {5,10,15,20, … } where 𝑁 is the set of
natural or counting number.
(iii) { states in Nigeria } constitute a set and Oyo state constitutes an element of the Set.
⇒ A universal set in any gwen context is the set of all elements under considerator at a
particular time. Itis denoted by 𝑈. A set which contains only one element. is called a unit
set or singleton, while a set which has no element is said to be emply or null set denoted
by 𝜙 or {} .
⇒ Subset and superset ide say that the set 𝐴 is a subset of the set 𝐵, denoted by 𝐴 ∈ 𝐵 if
and only if every element of ⇒ Two sets 𝐴 and 𝐵 are said to be comparable. If eithe 𝐴 ⊆
𝐵 or 𝐵 ⊆ 𝐴.
⇒ Equality of sets. Two non-empty sets 𝐴 and 𝐵 are sard to be equal of they both contan
the same elements, thatis 𝐴 ∈ 𝐵, 𝐵 ⊆ 𝐴 ⇒ 𝐴 = 𝐵. e.g the set = {−2,2} and 𝑇 = { The Set
of the quadratic equation 𝑥 2 − 4 = 0}-Then 𝑆 ⊆ 𝑇, 𝑇 ⊆ 𝑆 Implies 𝑆 = 𝑇
The number of elements in a given set is called the cardinality of the set. It is denoted by
𝑛(𝐴).

,⇒ 𝐴 set is said to be finite if it consists of exactly 𝑛 −different some positive integer;
otherwise it is infinite. An empty set has no elements, yet it is a finite set.

⇒ power set: Let 𝑆 be a non enpty fintie set Then the power bet of quantity of all possible
subsits of 𝑆. It is denoted by 𝑃(𝑠): ∣ 𝐶(𝑠) = 2|𝑠|
e.g 𝑆 = {𝑥, 𝑦, 𝑧}
𝑃(𝑠) = {𝜙, 𝑆, {𝑥 2 }, {𝑦}, {𝑧}, {𝑥, 𝑦}, {𝑥, 𝑧}, {𝑦, 𝑧}},
|𝑃(𝑠)| = 23 = 8.
⇒ Complement: Given 𝐴 ⊂ 𝑈, then the set of elements in 𝑈, Which are not in 𝐴 is the
complement of 𝐴. Itis denoted by 𝐴′ or 𝐴𝐶

Ac  { x : x  , x  A}



⇒ Difference of two sets:

The difference of two sets 𝐴 and 𝐵 denoted by 𝐴 − 𝐵 or 𝐴(𝐵. the elements in 𝐴 which
are not in 𝐴 − 𝐵 = {𝑥: 𝑥 ∈ 𝐴, 𝑥 ∈ 𝐵} by 𝐴 − 𝐵 or 𝐴 ∖ 𝐵 is the elents in 𝐴 which are not in
𝐵
𝐴−𝐵 = {𝑥: 𝑥 ∈ 𝐴, 𝑥 ∉ 𝐵}
e.g. 𝐴 = {1,2,3,4,5,6,7,8
𝐴 = {1,2,3,4,5,6,7,8,9,10}, 𝐵 = {2,5,7,1021,12,13}
𝐴 − 𝐵 = {1,3,4,6,8,9}
𝐵 − 𝐴 = {1,12,13}
⇒ The symmetric Difference
The symmetric difference of two sets 𝐴 and 𝐵 denoted by 𝐴 △ 𝐵 is defined as follows:

𝐴 △ 𝐵 = {𝑥: 𝑥 ∈ (𝐴 − 𝐵) or 𝑥 ∈ 𝐵 − 𝐴)}
= {𝑥: 𝑥 ∈ (𝐴 − 𝐵) or 𝑥 ∈ (𝐵 − 𝐴)}
= (𝐴 − 𝐵) ∪ (𝐵 − 𝐴)
= (𝐴 − 𝐵) ∪ (𝐵 − 𝐴) = (𝐴 ∪ 𝐵) − (𝐴 ∩ 𝐵)
𝐸 ⋅ 𝑔𝐴Δ𝐵 = (𝐴 ∪ 𝐵) − (𝐴 ∩ 𝐵)
𝐴Δ𝐵 = (𝐴 − 𝐵) ∪ (𝐵 − 𝐴)

, = {1,3,4,6,8,9} ∪ {16,12,13} = {1,3,4,6,8,9,11,12,13}

Bastc set Operations

Union of Sets

Union of two non-empty sets 𝐴 and 𝐵 denoted by 𝐴 ∪ 𝐵, I the set of all elenents which
belong to at least one of the two sets. Thus, 𝐴 ∪ 𝐵 = {𝑥: 𝑥 ∈ 𝐴 or 𝑥 ∈ 𝐵}
example: (1) et 𝑆 = {𝑥: 𝑥 2 + 7𝑥 + 12 = 0, 𝑥 ∈ 𝑍} and 𝑇 = {2,4,6,8,10}
then 𝑆𝑈𝑇 = {−4, −3,2,4,6,8,10}
(ii) 𝐴1 = {𝑥: 𝑥 2 + 7𝑥 + 12 = 0, 𝑥 ∈ ℤ}

𝐴2 = {2,4,6,8,10}
𝐴3 = {−3, −2, −1,0,1,2}
𝐴4 = {10,20,30}

Then 𝐴1 ∪ 𝐴2 ∪ 𝐴3 ∪ 𝐴4 is conventionally witthen as

4𝐴

⋃ 𝐴𝑖 = {−4, −3, −2, −1,0,1,2,6,8,10,20,30}.
𝑖=1


Intersection of two sets

Definition: The intersection of two sets 𝐴 and 𝐵 denoted by 𝐴 ∩ 𝐵 is the set of elements
common to both Aand 𝐵. 𝐴 ∩ 𝐵 = {𝑥: 𝑥 ∈ 𝐴 and 𝑥 ∈ 𝐵}. Clearly, 𝐴 ∩ 𝐵 = 𝐵 ∩ 𝐴. If 𝐴 ∩
𝐵 = then there is no element common to both Arand 𝐵𝑆 sets and 𝐴, 𝐵 are disjoint sets.
𝐴⋅𝑔 let 𝐴 = {2,6,10}, 𝐵 = {1,2,3,4,5}
𝐴𝑛𝐵 = {}
4

∏ 𝐴𝑖 = {} or 𝜙
i=1


Complements

Definition. The relative complement of B in A (also called the set difference) is the set: