introduction

In the realm of mathematics, three foundational concepts govern algebraic manipulation: Binary operations, axioms, and algebraic structures. Binary operations combine elements, axioms provide fundamental truths, and algebraic structures form the backbone of mathematical systems. Together, they lay the groundwork for exploring diverse mathematical theories and applications, such as number theory, vectors, complex numbers, and more. These concepts serve as the bedrock upon which various mathematical disciplines are built, enabling the exploration and understanding of complex mathematical phenomena.

Note: These three concepts form the foundational logic behind various theories in mathematics.

In arithmetic, we’re familiar with four primary operation: addition, subtraction, multiplication, and division i.e.\(\displaystyle \left(+,\;-,\;\times,\;\div\right)\) . However, fundamentally, there are only two operations: addition and multiplication. These operations, termed binary operations, and they form the cornerstone for fundamental counting principle symbolically the binary operation denote by ‘∗‘

Let’s delve into how subtraction and division relate to addition and multiplication, respectively.

Subtraction=> It essentially entails adding negative integers. For instance, while we write 10 – 5 = 5, in essence, it’s 10 + (-5) = 5

Division=> Similarly, division is akin to multiplying by reciprocal non-zero rational numbers, typically expressed as fractions. For instance, \(\displaystyle \frac{10}5=2\) can be understood as \(\displaystyle 10\times\left(\frac15\right)=2\)

note: when two numbers from a specific set undergo a binary operation, if the result remains within the same set, that set is said to be closed under the operation. This property is known as the closure property.

symbolically, this can be represented as  ƒ : Ζ * Ζ → Ζ   where, Z represents the set of integers.

Axioms

Axioms serve as the foundational properties, statements, or principles for an algebraic structure. The entire mathematical system is defined based on these axioms. They lack mathematical proof within the system they define because one cannot prove the consistency of a mathematical system while being within that system (this is a concept at the next level). Their acceptance without proof is a fundamental aspect of mathematical reasoning. There are two main types of axioms based on binary operations, which are as follows:

Addition axioms

1. Closure Property => If the sum of any two elements from a set, falls within that same set, known as the closure property of addition. For example, if a, b ∈ Ζ then a + b ∈ Ζ.
2. Additive Identity => If the sum of any two elements from a set equals one of the original elements, then the property is known as additive identity. For example if a ∈ Ζ than their exists an integer 0 such that a + 0 = a & 0 + a = a are the same. Here, 0 is known as the additive identity.
3. Additive Inverse => If the sum of any two elements from a set zero or additive identity, then those two elements are inverses of each other. This property is known as additive inverse. For example, if a ∈ Ζ, then their exists an inverse number – a ∈ Ζ such that a + (-a) = 0 & -a + a = 0 are same. Here,-a is known as the additive inverse or negative of ‘a’ and vice versa.
4. Associative Property =>The sum of any two particular elements out of three elements from a set is always equal if the order of elements doesn’t change. This property is known as the associative property of addition. For example, if a, b, c ∈ Ζ, than a + (b + c) =  (a + b) + c  but,  a + (b + c) ≠ (a + c) + b because the order of element is changed. This property is important for Group Theory.
5. Commutative Property => If the sum of any two elements from a set remains the same regardless of the order in which they are added, it is known as the commutative property of addition. For example, if a, b ∈ Ζ, than the order a + b or b + a results in the same value, i.e. a + b = b + a

Multiplicative Axioms

1. Closure Property => If the product of any two elements from a set, falls within that same set, known as closure property of multiplication. For example, if a, b ∈ Ζ then a × b ∈ Ζ.
2. Multiplicative Identity => If the product of any two elements from a set equals one of original elements, then the property is known as multiplicative identity. For example, if a ∈ Ζ than their exists an integer 1 such that a × 1 = a & 1 × a = a are the same. Here, 1 is known as the multiplicative identity.
3. Multiplicative Inverse => If the product of any two elements from a set one (1), then those two elements are inverse of each other. This property is known as multiplicative inverse. For example, if a ∈ R, than their exists inverse number \(\mathrm a^{-1}\) ∈ R such that: a × \(\mathrm a^{-1}\) = 1 & \(\mathrm a^{-1}\) × a = 1 are same. Here, \(\mathrm a^{-1}\) is known as the multiplicative inverse or reciprocal of ‘a’ and vice versa.
4. Associative Property =>The product of any two particular elements out of three elements from a set is always equal if the order of elements doesn’t change. This property is known as associativity property of multiplication. For example, if a, b, c ∈ Ζ than a × (b × c) = (a × b) × c  but, a × (b × c) ≠ (a × c) × b because the order of element is changed. This property is important for Group Theory.
5. Commutative Property => If the product of any two elements from a set remains the same regardless of the order they are multiply, it is known as commutative property of multiplication. For example, if a, b ∈ Ζ, than the order a × b or b × a results in the same value, i.e. a × b = b × a

The final property, known as the Distributive property, was not mentioned previously because this property does not belong to either addition or multiplication axioms. instead, it serves as a combined version of these axioms. In essence, if a, b, and c are elements from the set of integers (Z), then the product of ‘a’ and (b + c) equals the sum of the products ab and ac:

in equation form: a(b + c) = ab + ac

The final property, known as the Distributive property, was not mentioned previously because this property does not belong to either addition or multiplication axioms. instead, it serves as a combined version of these axioms. In essence, if a, b, and c are elements from the set of integers (Z), then the product of ‘a’ and (b + c) equals the sum of the products ab and ac:

in equation form: a(b + c) = ab + ac

Algebraic Structure

An algebraic structure serves as the fundamental framework upon which an entire mathematical system is constructed. Specifically, it is a set equipped with one or more binary operations that adhere to specific properties or axioms associated with their respective operations.

For example, consider the set of integers (Z) equipped with addition operation (+), denoted as (Z, +), which is an algebraic structure because the set ‘Z’ satisfies all the properties of the addition axiom. However, upon examining the multiplication axiom, the set ‘Z’ fails to satisfy the requirements of its respective operation, indicating that the outcome of that specific axiom does not belong within the set ‘Z’.

There exist five common properties shared by both operations, namely:.

(1). closed or closure.

(2). Identity element.

(3). Inverse element.

(4).commutative.

(5). associative.

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