Question

What is the difference between an axiom and a postulate?

Answer 1

Hint: First you need to define both the terms, axiom and postulates. Examples of both can be stated. The main difference is between their application in specific fields in mathematics.

Complete Step-by-Step solution:
We need to explain the difference between an axiom and a postulate.
First let us define an axiom.
An axiom is a statement or proposition which is regarded as being established, accepted, or self-evidently true on which an abstractly defined structure is based.
More precisely an axiom is a statement that is self-evident without any proof which is a starting point for further reasoning and arguments.
An axiom could be that $a + b = b + a$ for any two numbers $a$ and $b$.
Earlier axioms were considered different from postulates.
Postulate verbally means a fact, or truth of (something) as a basis for reasoning, discussion, or belief.
Postulates are the basic structure from which lemmas and theorems are derived.
For example, Euclid’s five postulates:
1. A straight line segment can be drawn joining any two points.
2. Any straight-line segment can be extended indefinitely in a straight line.
3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as centre.
4. All Right Angles are congruent.
5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two Right Angles, then the two lines inevitably must intersect each other on that side if extended far enough. This postulate is equivalent to what is known as the Parallel Postulate.
Nowadays ‘axiom' and 'postulate' are usually interchangeable terms.
One key difference between them is that postulates are true assumptions that are specific to geometry. Axioms are true assumptions used throughout mathematics and not specifically linked to geometry.

Note: A postulate is a pretty-good seeming often anecdotally confirmed assumption or hypothesis that lacks rigorous empirical or logical support or proof. An axiom can just be a postulate but, at least within logic, usually involves also being a tautology. Sometimes assertions that were long-held to be axioms ended up only really being postulates