What's the difference between a postulate and a theorem?

A postulate and a theorem are two fundamental concepts in mathematics that are used to prove and establish mathematical statements. Let's discuss their differences:

1. Definition:
- Postulate: Also known as an axiom, a postulate is a statement that is taken to be true without proof. It serves as a starting point for constructing a mathematical theory or a system. Postulates are basic assumptions that are universally accepted as true and are used as the foundation for proving other mathematical statements.
- Theorem: A theorem, on the other hand, is a statement that has been proven using logical reasoning and previously established postulates, theorems, and definitions. Theorems are derived from the postulates and are considered to be true based on the logical deductions made.

2. Role and Purpose:
- Postulate: Postulates are fundamental principles or rules that are assumed to be true in a given mathematical system or theory. They are used to establish the necessary starting point for building a mathematical framework. Postulates help define the properties and relationships within a mathematical structure.
- Theorem: Theorems are mathematical statements that have been proven using deductive reasoning. They provide a deeper understanding of mathematical concepts by demonstrating logical connections between previously established mathematical statements. Theorems are used to expand mathematical knowledge and establish new relationships and properties.

3. Proof:
- Postulate: Postulates are considered to be self-evidently true and do not require proof. They are accepted as given, serving as the foundational assumptions for further mathematical investigations.
- Theorem: Theorems require a logical proof using the rules of inference and previously accepted mathematical statements. A proof is a series of steps or arguments that logically demonstrate the truth of a theorem. Proofs provide a rigorous justification for the validity of the theorem.

In summary, while postulates serve as the foundational assumptions in mathematics, theorems are the proven statements derived from postulates and previous theorems. Postulates are taken as true without proof, whereas theorems require proof based on logical reasoning. Theorems build upon postulates and contribute to the development and expansion of mathematical knowledge.

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