Abstract Algebra Preliminaries and Basic Concepts

Abstract Algebra / Saturday, September 14th, 2019

Abstract Algebra Preliminaries and Basic Concepts (Sets, Relations, Mappings, Operations)

The study of Abstract Algebra or algebraic systems will require clear concepts on sets, relations, mappings, operations etc. and a quick review will help better understanding of the materials presented in this post series.

Sets

A set is a well-defined collection of objects. By well-defined collection of objects we understand that if S is a set and ‘a’ is some object, then either ‘a’ is definitely in S, denoted by a ∈ S, or a is definitely not in S, denoted by a ∉ S.

All the Essential Sets:

is the set of all natural numbers
is the set of integers
Z is the set of all positive integers
Z–  is the set of all negative integers
is the set of all rational numbers
is the set of all real numbers
C  is the set of all complex numbers

Subset

Let S be a set. A set T is said to be a subset of S if x ∈ T ⇒ x ∈ S. This means that each element of T is an element of S. And it is written as ST.

Null Set

The set containing no element is called the null set or the empty set or the void set. A null set is denoted by Ø.

Important Note

1. Every set is a subset of itself.
2. Ø is the subset of every set.
3. A set having n number of elements can have 2n number of subsets.

Cardinality

The cardinality of a finite set is defined to be the number of elements in the set. The cardinality of the empty set is 0.

Finite and Infinite Set

A set is said to be a finite set if either it is empty or it contains a finite number of elements, otherwise it is said to be an infinite set.

Equal Set

Two sets S and T are said to be equal if S is a subset of T and T is a subset of S.

Algebraic Operations on Set

1. The union of two sets A and B, written as AB, is the set {x | x ∈ A or x ∈ B}
2. The intersection of two sets A and B, written as AB, is the set {x | x ∈ A and x ∈ B}
3. Two sets A and B are said to be disjoint if AB = Ø.
4. Given two sets A and B, the difference AB is the set {x ∈ A | x ∉ B}.
5. The complement of a subset A is a subset of ξ, denoted by A’ or Ac and is defined by {x ∈ ξ | x ∉ A}

Properties of Union and Intersection

1. Consistency Property

The three relations BA, A B = A and AB = B are mutually equivalent. The following properties can be easily deduced from consistency property:

i) AØ = A, AØ = ØØA
ii) Aξ = ξ, Aξ = AAξ
iii) AA = A, AA = AAA Idempotent Property
iv) A ⋃ (AB) = A, A ⋂ (AB) = A Absorption Property

i) AB = BA
ii) AB = BA

3. Associative Property

i) A ⋃ (BC) = (AB) ⋃ C
ii) A ⋂ (BC) = (AB) ⋂ C

4. Distributive Property

i) A ⋃ (BC) = (AB) ⋂ (AC)
ii) A ⋂ (BC) = (AB) ⋃ (AC)

Cartesian product of two sets

If A and B are two sets, then the Cartesian product of the sets A and B, denoted by A x B, is the set A x B = {(x, y) | x ∈ A, x ∈ B}.
Thus, if A = {x, y} and B = {a, b, c}, then A x B is the set of distinct ordered pairs
{(x, a), (x, b) , (x, c) , (y, a) , (y, b) , (y, c)}

Relations

Let A and B be two sets and let ρ be a subset of A x B. Then ρ is called a relation from A to B. If (x, y) ∈ ρ, then x is said to be in relation ρ to y, written xρy. A relation from A to A is called a relation on A (or in A).

Let ρ be a relation in the set A. ρ is said to be

1. Reflexive if xρx for all x ∈ A.

2. Symmetric if xρy ⇒ xρx; x, y ∈ A.

3. Antisymmetric if xρy and xρx ⇒ x =y; x, y ∈ A.

4. Transitive if xρy and yρz ⇒ xρz; x, y, z ∈ A.

If the relation ρ is reflexive, symmetric and transitive then ρ is called an equivalence relation on A. If ρ is reflexive, antisymmetric and transitive then ρ is called a partial ordering relation on A.

An equivalence relation ρ defined on a set A partitions the set A into a number of disjoint classes, called the equivalence classes. Thus if a ∈ A, the equivalence classes of a is denoted by cl(a) and it is the set {x | aρx}.

Mappings

If S and T are non-empty sets, then a mapping from S to T is a subset M of S x T such that for every s ∈ S, there is a unique t ∈ T such that the ordered pair (s, t) is in M. Let σ be a mapping from S to T; we often denote this by writing σ: ST. If t is the image of s under σ, then we shall write t = σ(s).
Let S be any set. Let us define the mapping I: SS by I(s) = s for any s ∈ S. This mapping I is called identity mapping of S.

The mapping σ of S to T is said to be onto (or surjective) mapping if given t ∈ T there exists an element s ∈ S such that σ(s) = t.
The mapping σ of S to T is said to one-to-one (or injective) mapping if whenever s1 ≠ s2, (s1, s2S), then σ(s1) ≠ σ(s2).

A one-to-one and onto mapping is called a bijective mapping.
The two mapping σ and τ of S into T are said to be equal if σ(s) = τ(s) for every s ∈ S.
If σ: ST and τ: TU, then the composition of σ and τ (also called their product) is the mapping τoσ: SU defined by means of (τoσ)(s) = τ(σ(s)) for every s ∈ S.

Thus for the composition τoσ of σ and τ, we shall say always mean: first apply σ and then τ.

 Lemma 1.1 (Associative Law)

If σ: ST, τ: TU and µ: UV are three mappings, then the associative law (µoτ)oσ = µooσ) holds.

Binary operation

A binary operation o on a set S is a rule that assigns to each ordered pair of elements of the set S some element of the set. Thus, for an arbitrary set S, we call a mapping of S x S into S, a binary operation of S.

 Example 01

On Z+ (the set of positive integers), define a binary operation o by aob equals the smaller of a and b or the common value if a = b; a, b ∈ Z+.

Thus 2o11 = 2; 15o10 = 10 and 4o4 = 4.

 Example 02

On Z+ define the operation o by aob = a/b, a, b ∈ Z+. Clearly the operation o is not a binary operation as 1o3 = 1/3 ∉ Z+.

A binary operation o on a set S is commutative if and only if aob = boa for all a, b ∈ S. The operation o is associative if and only if (aob)oc = ao(boc) for all a, b, c ∈ S.

 Remark

In defining a binary operation on a set S it is necessary that (i) exactly one element is assigned to each ordered pair (a, b), (a, b ∈ S) and (ii) for each ordered pair of elements of S, the element assigned to it again in S. If the second condition is not satisfied then we say that S is not closed under the operation.