Mathematical Equality: Fundamentals and Applications

By Fouad Sabry

What Is Mathematical Equality

In the field of mathematics, equality refers to a relationship that exists between two numbers or, more generally speaking, two mathematical expressions. This relationship asserts that the quantities share the same value or that the expressions reflect the same mathematical object. The statement that A and B are equal can be written as "A equals B" and spoken as "A is equal to B." An "equals sign" is the name given to the symbol "=". Two things are considered to be separate if they cannot be compared to one another.

How You Will Benefit

(I) Insights, and validations about the following topics:

Chapter 1: Equality (mathematics)

Chapter 2: Equivalence relation

Chapter 3: Equivalence class

Chapter 4: First-order logic

Chapter 5: Groupoid

Chapter 6: Isomorphism

Chapter 7: Peano axioms

Chapter 8: Algebraic structure

Chapter 9: Reflexive relation

Chapter 10: Transitive relation

(II) Answering the public top questions about mathematical equality.

(III) Real world examples for the usage of mathematical equality in many fields.

(IV) 17 appendices to explain, briefly, 266 emerging technologies in each industry to have 360-degree full understanding of mathematical equality' technologies.

Who This Book Is For

Professionals, undergraduate and graduate students, enthusiasts, hobbyists, and those who want to go beyond basic knowledge or information for any kind of mathematical equality.

• Language: English
• Publisher: One Billion Knowledgeable
• Release date: Jun 25, 2023

Book preview

Chapter 1: Equality (mathematics)

To state that two numbers have the same value or that two mathematical expressions describe the same mathematical object is what we mean by equality in mathematics. When A and B are equal to one another, we write A = B and say A equals B. An equals sign is the symbol =. It is argued that two things are different if they cannot be equated.

For example:

x=y means that x and y denote the same object.

The identity {\displaystyle (x+1)^{2}=x^{2}+2x+1} means that if x is any number, Consequently, the two phrases are equivalent.

One possible interpretation is that the equals sign is saying that the two sides do the same operation.

{\displaystyle \{x\mid P(x)\}=\{x\mid Q(x)\}} if and only if {\displaystyle P(x)\Leftrightarrow Q(x).} This assertion, whose nomenclature is based on set construction, means that if the elements satisfying the property P(x) are the same as the elements satisfying {\displaystyle Q(x),} then the two uses of the set-builder notation define the same set.

When two sets share the same elements, we say that they are equal. This is a standard axiom in the field of set theory, Axiom of Extensibility, for short.

The etymology of the word is from the Latin aequālis (equal, like, comparable, similar) from aequus (equal, level, fair, just).

The property of equivalence through substitution states that for any two numbers a and b and any expression F(x), if a = b then F(a) = F(b) (provided that both sides are well-formed).

Here are a few concrete cases in point::

The function F(x) is equal to x + c if and only if the real numbers a and b are equal; In this case, F(x) is equal to xc, where a, b, and c are all real values; If a=b, then ac=bc for every a, b, and c in the real numbers (here, F(x) is xc); F(x) is x/c if and only if (for any real numbers a, b, and c) (a=b) and (c>0).

Property of self-equivalence: for any number a, a = a.

The property of symmetry states that if a = b, then b = a for any quantities a and b.

For any three quantities a, b, and c, the transitive property states that if a = b and b = c, then a = c.

Equality is an equivalence relation because of the last three features listed. They were first established as part of the Peano axioms for numbers in general. Despite widespread belief to the contrary, the symmetric and transitive qualities can be derived from the substitution and reflexive ones.

When both A and B lack complete specification or are dependent on other variables, equality becomes a statement that can be true for certain values but false for others. The truth value (true or false) of the relation of equality is determined by its two arguments. It is called comparison when calculated in computer programming from two expressions.

When both A and B could be functions of a few independent variables, Consequently, if A=B, then both A and B define the same function.

An identity is a term used to describe this kind of equality between functions.

An example is

{\displaystyle \left(x+1\right)\left(x+1\right)=x^{2}+2x+1.}

Sometimes, on the other hand, an identity is written with a triple bar:

{\displaystyle \left(x+1\right)\left(x+1\right)\equiv x^{2}+2x+1.}

Obtaining values for a set of variables is the problem represented by an equation, called unknowns, which satisfy the given equality.

In some contexts, the word equation might mean an equality relation that holds true solely for the values of the variables in question.

For example, x^2 + y^2 = 1 is the equation of the unit circle.

One must infer the correct interpretation based on the semantics of expressions and the context, as there is no standard notation that differentiates between an equation and an identity or other application of the equality relation. For all possible values of a set of variables, an identity is claimed to hold. The term equation is sometimes used to indicate identity, but more commonly it refers to a statement about which parts of the variable space satisfy the statement.

Some logical frameworks do not recognize the concept of equality. This represents the fact that formulas including integers, the four basic arithmetic operations, the logarithm, and the exponential function are unable to determine whether or not two real numbers are equal. That is to say, there is no algorithm that can determine whether or not two things are equal.

The binary relation is approximately equal (denoted by the symbol \approx ) between real numbers or other things, Regardless of how well specified it is, has no connecting verb (since many small differences can add up to something big).

However, Generally speaking, equality is a transitive concept.

A questionable equality under test may be denoted using the ≟ symbol.

Regarded as a connection,, Equality is the prototypical equivalence relation on a set, which are the reflexive binary relations, Parallel and flowing.

One type of equivalence connection is the identity relation.

Conversely, An equivalency relation, denoted by R, and let us denote by xR the equivalence class of x, consisting of all z's where x R z.

Then the relation x R y is equivalent with the equality xR = yR.

Since the equivalence classes of equality are always minimal, it follows that equality is the best equivalence relation on any set S. (every class is reduced to a single element).

In certain situations, There is a clear difference between equality and equivalence or isomorphism.

For example, fractions and rational numbers can be distinguished, the latter being equivalence classes of fractions: the fractions 1/2 and 2/4 are distinct as fractions (as different strings of symbols) but they represent the same rational number (the same point on a number line).

As a result of this differentiation, the concept of a quotient set is introduced.

Just like the sets, {\displaystyle \{{\text{A}},{\text{B}},{\text{C}}\}} and {\displaystyle \{1,2,3\}}

are not equivalent because one contains letters and the other has numbers, but they are isomorphic because they both include three elements and may be transformed into each other via a bijection because they have the same shape. For Instance

\text{A} \mapsto 1, \text{B} \mapsto 2, \text{C} \mapsto 3.

Isomorphism can also be accomplished in a

\text{A} \mapsto 3, \text{B} \mapsto 2, \text{C} \mapsto 1,

Furthermore, any statement identifying these sets depends on choice of identification, requiring the reader to make a decision in order to do so. The difference between equality and isomorphism is crucial to category theory and was a driving force in its development.

Some situations, Depending on what features and structures are being compared, two mathematical objects may be deemed equivalent.

The word congruence (and the associated symbol \cong ) is frequently used for this kind of equality, and is characterized as the intersection of all classes of isomorphism among the items.

Take geometry as an example, When two geometric shapes can be brought into perfect alignment with each other, we say that they are congruent, classes of isometries between shapes can be described by the equality/congruence relation.

Isomorphisms of sets are an analogous concept, The creation of category theory was inspired, in part, by the need to distinguish between isomorphisms and equality/congruence between such property- and structure-bearing mathematical objects, as well as in univalent bases and homotopy type theory.

Leibniz gave the following definition of equality:

If you give me an x and a y, and I provide you a predicate P, then if and only if P(x) then and only if x = y. (y).

In set theory, there are two axiomatizations of set equality, one for when equality is present in the underlying first-order language and one for when it is not.

The axiom of extensionality states that any two sets sharing the same elements are the same set in first-order logic with equality.

Logic axiom: x = y ⇒ ∀z, (z ∈ x ⇔ z ∈ y)

Logic axiom: x = y ⇒ ∀z, (x ∈ z ⇔ y ∈ z)

Set theory axiom: (∀z, (z ∈ x ⇔ z ∈ y)) ⇒ x = y

It may be seen as a matter of convenience to include around 50% of the effort into first-order reasoning, as noted by Lévy.

The responsibility of establishing equality and demonstrating all its features has been lifted from the logic's shoulders, which is why first-order predicate calculus is taken up with equality.

Two sets are considered equal if they have the same elements, according to the definition used in first-order logic without equality. The axiom of extensionality then stipulates that any two sets of equal size are contained in each other.

Set theory definition: x = y means ∀z, (z ∈ x ⇔ z ∈ y)

Set theory axiom: x = y ⇒ ∀z, (x ∈ z ⇔ y ∈ z)

{End Chapter 1}

Chapter 2: Equivalence relation

An equivalency relation is a type of binary relation that satisfies the mathematical definitions of being reflexive, symmetric, and transitive. One typical type of equivalency relation is the equipollence relation between line segments in geometry.

Each equivalence relation distinguishes distinct equivalence classes within the underlying set. It is only if they have the same equivalence class that two items of the given set can be considered equal to one another.

Various notations are used in the literature to denote that two elements a and b of a set are equivalent with respect to an equivalence relation {\displaystyle R;} the most common are a\sim b and a ≡ b, which are used when R is implicit, and variations of {\displaystyle a\sim _{R}b} , a ≡R b, or {\displaystyle {a\mathop {R} b}} to specify R explicitly.

Non-equivalence may be written a ≁ b or a\not \equiv b .

A binary relation \,\sim\, on a set X is said to be an equivalence relation, Specifically, if the action is reflexive, Transitive and symmetric.

That is, for all a, b, and c in {\displaystyle X:}

{\displaystyle a\sim a} (reflexivity).

a\sim b if and only if {\displaystyle b\sim a} (symmetry).

If a\sim b and {\displaystyle b\sim c} then {\displaystyle a\sim c} (transitivity).

X together with the relation \,\sim\, is called a setoid.

The equivalence class of a under {\displaystyle \,\sim ,} denoted {\displaystyle [a],} is defined as {\displaystyle [a]=\{x\in X:x\sim a\}.}

Algebraic relations, if R\subseteq X\times Y and S\subseteq Y\times Z are relations, then the composite relation {\displaystyle SR\subseteq X\times Z} is defined so that {\displaystyle x\,SR\,z} if and only if there is a y\in Y such