Real Analysis/Abstract Algebra Basics

In The Real Numbers section, many of the types of numbers familiar in elementary mathematics—for example the integers or the rational numbers—are often described with certain properties, such as obeying the commutative law or associative law. These properties are often described as is in Real Analysis are are not often mentioned any further as these topics will often fall out of the scope of Real Analysis. Thus, this page is completely option to those solely studying Real Analysis. However, these terms are the foundation of a greater field of mathematics that may be of interest to your mathematical journey. Thus, this section will illustrate just the basics of the types of algebraic structures discussed in Abstract Algebra as they are applied to the familiar sets of numbers in elementary mathematics.

Note that the following page will not be necessarily rigorous or free from possible notation abuse! This is simply to provide a primer to the topic. For those who wish to read more on Abstract Algebra, the following link w:Abstract Algebra will take you to the Wikipedia page and the wikibook Abstract Algebra will discuss the topic at greater detail.

Definition

Abstract Algebra can be thought of as the mathematical field of studying the operations of algebra, much like how Analysis can similarly be thought of as a field in which one studies the limit. Likewise, Abstract Algebra can be imagined as a study of the relationship between algebraic properties in regards to algebraic structures, whereas Analysis can be imagined as a study of the following concepts consequentially derived through the introduction of a limit. As such, the fundamental building blocks of Abstract Algebra are the following

Concept of an Algebraic Structure

---

A set $A$ and some operation $\ast$ that satisfies some axiomatic property with all elements $a$ in $A$.

Operations are the symbols in which one or more inputs are "transferred" to some output. Similar to how limits compute certain values and derivatives and integrals transform equations into other equations, operations transform inputs into some other output. Abstract Algebra seeks to create theorems for certain kinds of operation "transformations" in a general sense much like how we use variables to substitute values in Analysis.

In Real Analysis, the following sets are often used:

• The natural numbers $\mathbb {N}$
• The integers $\mathbb {Z}$
• The rational numbers $\mathbb {Q}$
• The real numbers $\mathbb {R}$

and the following operations are usually a given for them:

• Addition $+$
• Subtraction $-$
• Multiplication $\times$ or $\cdot$ or the absence of a space between variables
• Division $\div$ or $^{-1}$ or the rational function notation
• Exponentiation $a^{x}$

with their various methods of computation either defined in their respective sections or assumed to be known through elementary mathematics.

One may notice that the first section of this wikibook is to essentially define these sets, operations, and the operations' properties and functionality as axiomatically true in order to rapidly move on to the Analysis field of mathematics. This is a usual requirement to start off higher mathematics at in order to provide rigor to the calculus, polynomials, and other such concepts for the mathematical objects usually ascribed to when people talk about mathematics. However, this section will take a detour from this and discuss these operations at a greater depth.

Notation

Abstract algebra use common notation forms that are relatively uniform throughout mathematics. The most notable facet in its notation though is its use of operator variables, which are commonly notated as either $\ast$, $\bullet$, or $\star$. In usual parlance, operation refers to the full definition; operator refers to the symbol used to denote the operation; operand (also called input) refers to the variables the operator will operate on; and arity refers to the number of operands used in the operator.

Succinctly written in mathematical notation, the operation $\ast$ is written as

$\ast :A\times B\times \cdots \times Z\rightarrow Y$

where the sets $A,B,\cdots ,Z$ represent the set that the operator operates on—$\times$ being the symbol to separate each operand, and the set $Y$ represents the codomain (also called range). A close look at the notation reveals that the notation for operators emulates the function definition. This is not done accidentally; operators are defined as functions.

Like functions, certain relationships between the sets define special properties of the operator. We will look at many of them later in #Algebraic Structure. However, two will be important to define a shorthand version that one will often see in higher mathematics. First, we will define two terms:

Definition of Closed

---

An operator $\ast$ is closed if all sets in its definition are equal. This is depicted as $\ast :A\times \cdots \times A\rightarrow A$.

This is usually written as "closed under X", where X is the name of the operator.

Definition of Binary Operator

---

An operator $\ast$ that takes two operands. This is depicted as $\ast :A\times B\rightarrow Y$

Since many common operators (and nearly all operators examined in Real Analysis) are binary and closed, there exists a special notation for these kinds of operators. For any given set $A$ and operator $\ast$ that operates on elements in the set $A$ and is also binary and closed, the following notation is used:

$(A,\ast )$

Almost all of the traditional operators are infix; the way to write the operation is by inserting the operator in between the operands. This is notated as $a\bullet b$ for some compatible set $A$ for the operator $\ast$ and elements $a,b\in A$. However, other notations include prefix, when the operator is written in front of its operand formally as $\ast (a,b)$, and postfix, when the operator is written behind its operands formally as $(a,b)\ast$. Prefix and postfix are usually used for unary operations (operations that only take 1 operand) and its brackets are often dropped.

Properties

The algebraic structures described in Real Analysis often work with the following mathematical properties. Their inclusion or exclusion is an essential component for differentiating different algebraic structures. A non-exhaustive list include:

• The Commutative Property: whether the position of the inputs affects the output
• The Associative Property: whether the order in which duplicate operations are evaluated affects the output
• The Distributive Property: whether an operation evaluating another operation's operands first affects the output
• Inverse elements: whether there exists an operand for any given input that outputs the identity
• Identity element: whether there exists an operand that returns the input as an output

other important properties is whether the above properties are left or right; whether the property is only expressed when the binary operand is on the left or right side.

With the inclusion of these properties, it ought to be noted that certain operators are often used to imply a certain kind of algebraic property. Common ones are the symbols $+$ and $\times$, which are used much in the same way to imply properties as the variables $x$, $y$, and $a$ are. To summarize, the symbols $+$ and $\times$ usually imply the properties expressed in addition and multiplication, respectively, and the most important property that is usually implied is the distributive property with the symbols acting as they usually do in arithmetic.

Algebraic Structures of Numbers

Note that the following algebraic structures are only mentioned in relationship to the sets of numbers found in this wikibook. A more exhaustive list can be found on the page w:Category:Algebraic structures and a more ordered list can be found on the page w:Algebraic structure.

Using abstract algebra, the algebra used in elementary mathematics on the various types of numbers and their operators becomes concrete examples of abstract algebra concepts like how polynomials are a concrete example of a function. This makes it very easy to translate algebraic theorems, such as the multiplication of two binomials, into generalized theorems that work for any system of variables that happen to have operators of certain properties.

To enumerate the following sets of numbers, we can categorize them in a way fit for abstract algebra:

Natural Numbers

The natural numbers $\mathbb {N}$ (including zero) are:

• Reflexive, Symmetric, and Transitive over equalities
• Transitive over inequalities

and are thus:

${\displaystyle }$