Table of Contents

# 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

**is the set of integers**

*Z***is the set of all positive integers**

*Z*^{+ }**is the set of all negative integers**

*Z*^{– }**is the set of all rational numbers**

*Q***is the set of all real numbers**

*R***is the set of all complex numbers**

*C*## 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 2^{n} 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 x*ρ*y. 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 x*ρ*x for all x ∈ *A*.

2. **Symmetric** if x*ρ*y ⇒ x*ρ*x; x, y ∈ *A*.

3. **Antisymmetric** if x*ρ*y and x*ρ*x ⇒ x =y; x, y ∈ *A*.

4. **Transitive** if x*ρ*y and y*ρ*z ⇒ x*ρ*z; 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 | a*ρ*x}.

## 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 s_{1} ≠ s_{2}, (s_{1}, s_{2} ∈ *S*), then σ(s_{1}) ≠ σ(s_{2}).

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_{o}σ: *S* → *U* defined by means of (τ_{o}σ)(s) = τ(σ(s)) for every s ∈ *S*.

Thus for the composition τ_{o}σ of σ and τ, we shall say always mean: first apply σ and then τ.

Lemma 1.1 (Associative Law) |

If σ: *S* → *T*, τ: *T* → *U* and µ: *U* → *V* are three mappings, then the associative law (µ_{o}τ)_{o}σ = µ_{o}(τ_{o}σ) 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 a

*o*b equals the smaller of a and b or the common value if a = b; a, b ∈

*Z*^{+}*.*

Thus 2*o*11 = 2; 15*o*10 = 10 and 4*o*4 = 4.

Example 02 |

On ** Z^{+} **define the operation

*o*by a

*o*b = a/b, a, b ∈

*Z*^{+}*.*Clearly the operation

*o*is not a binary operation as 1

*o*3 = 1/3 ∉

**.**

*Z*^{+}A binary operation *o* on a set *S* is commutative if and only if a*o*b = b*o*a for all a, b ∈ *S*. The operation *o* is associative if and only if (a*o*b)*o*c = a*o*(b*o*c) 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.;