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:

N   is the set of all natural numbers
Z   is the set of integers
Z⁺  is the set of all positive integers
Z^–  is the set of all negative integers
Q   is the set of all rational numbers
R   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 S⊆T.

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

Properties of Union and Intersection

1. Consistency Property

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

i) A ⋃ Ø = A, A ⋂ Ø = Ø ⇒ Ø ⊂ A
ii) A ⋃ ξ = ξ, A ⋂ ξ = A ⇒ A ⊂ ξ
iii) A ⋃ A = A, A ⋂ A = A ⇒ A ⊂ A Idempotent Property
iv) A ⋃ (A ⋂ B) = A, A ⋂ (A ⋃ B) = A Absorption Property

2. Commutative Property

i) A ⋃ B = B ⋃ A
ii) A ⋂ B = B ⋂ A

3. Associative Property

i) A ⋃ (B ⋃ C) = (A ⋃ B) ⋃ C
ii) A ⋂ (B ⋂ C) = (A ⋂ B) ⋂ C

4. Distributive Property

i) A ⋃ (B ⋂ C) = (A ⋃ B) ⋂ (A ⋃ C)
ii) A ⋂ (B ⋃ C) = (A ⋂ B) ⋃ (A ⋂ C)

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: S → T. 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: S → S 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, s₂ ∈ S), 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: S → T and τ: T → U, then the composition of σ and τ (also called their product) is the mapping τ_oσ: S → U defined by means of (τ_oσ)(s) = τ(σ(s)) for every s ∈ S.

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

If σ: S → T, τ: T → U and µ: U → V are three mappings, then the associative law (µ_oτ)_oσ = µ_o(τ_oσ) 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.

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.

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.

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.;