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Axioms and postulates are the basic assumptions and are accepted without demonstration. All other theorems must be proven with the aid of these basic assumptions. At the foundation of the various sciences lay certain additional hypotheses which were accepted without proof. Such a hypothesis was termed a postulate. While the axioms were common to many sciences, the postulates of each particular science were different. For Example:

**Axiom -** Things which are equal to the same thing are also equal to one another(common to many sciences).

**Postulate -** It is possible to draw a straight line from any point to any other point(particularly in Geometry).

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Axioms are definitons or statements which is accepted without any proof because it is impossible to prove them. These assumptions are actually 'obvious universal truths'. Axioms are assumptions used throughout mathematics and not only in geometry.

For eg: corresponding angles axiom, linear pair axiom, etc.

Postulates are statements or assumptions which can be only applied to geometry.

For eg: A straight line may be drawn from any one point to another.

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An axiom is a statement that everyone believes is true, such as "supply equals demand" or "the only constant is change." Mathematicians use axiom to refer to established proofs.

Many axioms are so widely used they become clichés—here's one to help you remember what the word means. Axioms are so widely accepted they're seen almost as facts. A police officer interrogating a witness might just as easily say, "Stick to the axioms," as "Stick to the facts, ma'am."

Assume something or present it as a fact and you postulate it. Physicists postulate the existence of parallel universes, which is a little mind-blowing.

Anyone who has suffered through geometry class is familiar with some of the greatest hits, like Euclid's postulate and the point-line-plane postulate. Those are propositions that have to be assumed for other mathematical statements to follow logically. As a verb (pronounced "POST-you-late") it describes the act of presenting an idea, theory, belief, or concept.

Based on logic, an axiom or postulate is a statement that is considered to be self-evident. Both axioms and postulates are assumed to be true without any proof or demonstration. Basically, something that is obvious or declared to be true and accepted but have no proof for that, is called an axiom or a postulate. Axioms and postulate serve as a basis for deducing other truths.

The ancient Greeks recognized the difference between these two concepts. Axioms are self-evident assumptions, which are common to all branches of science, while postulates are related to the particular science.

Axioms

Aristotle by himself used the term “axiom”, which comes from the Greek “axioma”, which means “to deem worth”, but also “to require”. Aristotle had some other names for axioms. He used to call them as “the common things” or “common opinions”. In Mathematics, Axioms can be categorized as “Logical axioms” and “Non-logical axioms”. Logical axioms are propositions or statements, which are considered as universally true. Non-logical axioms sometimes called postulates, define properties for the domain of specific mathematical theory, or logical statements, which are used in deduction to build mathematical theories. “Things which are equal to the same thing, are equal to one another” is an example for a well-known axiom laid down by Euclid.

Postulates

The term “postulate” is from the Latin “postular”, a verb which means “to demand”. The master demanded his pupils that they argue to certain statements upon which he could build. Unlike axioms, postulates aim to capture what is special about a particular structure. “It is possible to draw a straight line from any point to any other point”, “It is possible to produce a finite straight continuously in a straight line”, and “It is possible to describe a circle with any center and any radius” are few examples for postulates illustrated by Euclid.

An axiom is a statement, usually considered to be self-evident, that assumed to be true without proof. It is used as a starting point in mathematical proof for deducing other truths.

Classically, axioms were considered different from postulates. An axiom would refer to a self-evident assumption common to many areas of inquiry, while a postulate referred to a hypothesis specific to a certain line of inquiry, that was accepted without proof. As an example, in Euclid's Elements, you can compare "common notions" (axioms) with postulates.

In much of modern mathematics, however, there is generally no difference between what were classically referred to as "axioms" and "postulates". Modern mathematics distinguishes between logical axioms and non-logical axioms, with the latter sometimes being referred to as postulates.

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