28 November 2023

geometry is a branch of mathematics which deals with the study of shapes, sizes and spatial relationships between them. This field of study has existed for thousands of years and has contributed to the advancement of many formal sciences, such as architecture, engineering, and astronomy.

At the heart of geometry is a set of fundamental principles known as axioms, which serve as the basis for all proofs and constructions based on geometry. In this post we will explore the concept of axioms in geometry and its importance in the study of mathematics.

Axioms are self-evident truths that serve as starting point for mathematical reasoning. They are statements that are accepted as true without proof and are the basis of all mathematical proofs.

Definition and concept of an axiom in geometry

An axiom is a statement that it is assumed true, without the need for proof and that serves as a starting point for logical deductions and theorems. In other words, the axioms are the building blocks of geometry and provide a framework for logical reasoning and deduction.

the axioms are considered self-evident truths, which means that they cannot be deduced from any other cause or principle of geometry. Therefore, they constitute the basis of all logical conclusions and theorems that are derived in the field of geometry.

Understanding the nature and role of axioms in geometry is essential to develop a strong foundation in this field of mathematics.

What is an axiom used for in mathematics?

In the field of mathematics, an axiom is a statement that is accepted without proof. It is considered a basic and fundamental principle on which mathematical reasoning is based. Axioms are used to define the properties and relationships of mathematical objects and provide a starting point for proving other mathematical claims.

In geometry, the axioms are used to define the basics of points, lines, and planes, and provide the framework for developing geometric proofs. The use of axioms allows mathematicians to develop a logical and coherent system of mathematics, based on a set of agreed assumptions.

What is the origin and history of axioms in mathematics?

Axioms are fundamental principles of mathematics that are accepted without proof. The concept of an axiom dates back to ancient Greek philosophy, where philosophers such as Euclid used axioms and postulates in geometric proofs.

The Elements of Euclid, considered one of the works most influential in the history of mathematics, is a collection of geometric propositions that are largely based on axioms. Aristotle also used axioms in his analysis of logical propositions, which formed the basis of his Organon.

Throughout time, axioms have been a essential part of mathematical reasoning, and mathematicians continue to use them as a basis for developing new ideas and theories. Today, axioms are an integral part not only of geometry, but of many other branches of mathematics, such as set theory, analysis, and algebra.

What are the types of axioms in geometry?

in geometry there are several types of axioms, each of which has a specific function. The most commonly recognized type is the Euclidean axiom, which is a set of basic assumptions describing the properties of points, lines, and planes in Euclidean space.

These axioms lay the basics of geometry euclidean and include statements such as ‘two points determine a line’ and ‘a line can be extended infinitely in both directions’.

Another type of axiom is the projective axiom, which is based on properties of projective space rather than Euclidean space. The axioms of projective geometry apply to objects that exist within a single plane or higher-dimensional space, and are used to describe properties such as perspective and symmetry.

algebraic axioms

Algebraic axioms are a fundamental part of mathematics that provide a set of rules that govern the behavior of algebraic operations. In the context of geometry, axioms are statements that are taken for granted without the need for proof to define the basic properties of geometric objects.

Some algebraic axioms and their main characteristics are:

  • group axioms: Groups are algebraic structures that consist of a set of elements and a binary operation that fulfills certain properties. The group axioms are:
  1. to) lock: For all elements ‘a’ and ‘b’ in the group, the result of their operation is also in the group.
  2. b) associativity: For all elements ‘a’, ‘b’ and ‘c’ in the group, (a * b) * c = a * (b * c).
  3. c) Existence of the neutral element: There exists an element ‘e’ in the group such that, for every element ‘a’ in the group, a * e = e * a = a.
  4. d) Existence of the inverse: For every element ‘a’ in the group, there exists an element ‘b’ in the group such that a * b = b * a = e (neutral element).
  • ring axioms: Rings are algebraic structures that include two binary operations (usually addition and multiplication) that satisfy certain properties. The ring axioms include the group axioms for addition, plus:
  1. to) commutativity of addition: For all elements ‘a’ and ‘b’ in the ring, a + b = b + a.
  2. b) distributivity: For all elements ‘a’, ‘b’ and ‘c’ in the ring, a * (b + c) = a * b + a * c and (a + b) * c = a * c + b * c .
  3. c) Optionally, commutativity of multiplication: For all elements ‘a’ and ‘b’ in the ring, a * b = b * a. If this property is satisfied, the ring is called commutative.
  • field axioms: Fields are algebraic structures that combine the properties of groups and rings in both operations (addition and multiplication). The field axioms include the ring axioms, plus:
  1. to) Existence of the neutral element of multiplication: There exists an element ‘1’ in the field such that, for every element ‘a’ in the field, a * 1 = 1 * a = a.
  2. b) Existence of the multiplicative inverse: For every non-zero element ‘a’ in the field, there exists an element ‘b’ in the field such that a * b = b * a = 1 (neutral element of multiplication).

axioms of order

the axioms of order define the relationship between pointslines and planes in space, and allow us to draw logical conclusions about them.

Some axioms of order and its main features are:

  • reflexivity axiom: This axiom states that each element of an ordered set is related to itself. That is, for every element ‘a’ in the set, it is true that a ≤ a.
  • antisymmetry axiom: According to this axiom, if two elements of an ordered set are related in both ways, then they must be equal. That is, for all elements ‘a’ and ‘b’ in the set, if a ≤ b and b ≤ a, then a = b.
  • transitivity axiom: This axiom states that if an element ‘a’ is related to an element ‘b’, and ‘b’ is related to an element ‘c’, then ‘a’ must also be related to ‘c’. That is, for all elements ‘a’, ‘b’ and ‘c’ in the set, if a ≤ b and b ≤ c, then a ≤ c.
  • Axiom of totality (or trichotomy): This axiom ensures that, in an ordered set, any pair of distinct elements must be related in some sense. That is, for all the elements ‘a’ and ‘b’ in the set, it is true that a ≤ bob ≤ a.

topological axioms

The topological axioms are the foundations on which the topology is built. Topology is a branch of mathematics that deals with the study of spaces, their properties, and their relationships to each other.

Some of the most common topological axioms are:

  • Axioms of topological spaces: Topological spaces are sets endowed with a structure that allows defining concepts such as continuity, limit and convergence. The axioms that define a topological space are:

a) The empty set and the complete set are open.

b) The finite intersection of open sets is an open set.

c) The arbitrary union of open sets is an open set.

  • separation axioms: These axioms allow us to distinguish and classify topological spaces according to the way in which their elements can be separated. Some examples are:

to) Axiom T0 (Kolmogorov spaces): Given two distinct points in space, there exists an open set that contains exactly one of them. This axiom is used to study spaces in which it is possible to distinguish different points.

b) Axiom T1 (Fréchet spaces): Given two distinct points in space, there are disjoint open sets that contain each of them. This axiom is useful for studying spaces with a greater degree of separation between their points.

c) Axiom T2 (Hausdorff spaces): Given two distinct points in space, there are disjoint open sets that contain each of them. Hausdorff spaces are fundamental in mathematical analysis, since they guarantee the uniqueness of limits and convergence of sequences.

  • compactness axioms: Compactness is a topological property that generalizes the notion of a closed and bounded set in metric spaces. A topological space is compact if every open collection that covers it has a finite subset that also covers it. Compactness is important in theorems such as the Bolzano-Weierstrass Theorem, which states that every bounded infinite set in a metric space has at least one accumulation point.

Examples of the use of axioms in differential calculus

He diferential calculus It is a branch of mathematics that deals with the study of the rates of variation of functions. It uses a set of fundamental concepts and rules, among which are the axioms.

In differential calculus, axioms are essential tools that help establish the fundamentals of the subject. For example, the axiom of continuity is used to prove the existence of limits and the continuity of a function, while the axiom of differentiability helps determine whether a function is differentiable at a given point.

Other axioms, such as the mean value theorem, the intermediate value theorem, and the chain rule, they are also essential in the differential calculus. The use of axioms is necessary because calculus deals with infinitely small objects, so it is often difficult to prove claims using direct methods.

The use of axioms in differential calculus has proven to be a powerful tool for developing theories and understand the properties of functions.

Understanding the axioms of mathematics is essential to mastering the subject, since they constitute the foundation of mathematical proofs and reasoning. By knowing and applying these basic axioms, a deep understanding of the principles that govern the world around us can be developed.

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