Relation between Axiom and Theorem

Axiom and Theorem are two terms that are commonly used in mathematics. They are related to each other in various ways. In this article, we will discuss the relationship between axiom and theorem.

Definition of Axiom and Theorem

An axiom is a statement that is accepted as true without proof. It is a self-evident truth that is used as a starting point for reasoning. Axioms are also known as postulates.

A theorem, on the other hand, is a statement that is proven to be true based on other statements that have been previously proven. Theorems are derived from axioms.

Relationship between Axiom and Theorem

The relationship between axiom and theorem can be explained as follows:

1. Axioms are the foundation of mathematics:
Axioms are the basic assumptions upon which all mathematical reasoning is based. They are the starting point for any mathematical proof. Without axioms, it would be difficult to make any mathematical statements.

2. Theorems are derived from axioms:
Theorems are statements that are proven based on previously proven statements. These previously proven statements are often axioms. Therefore, theorems are derived from axioms.

3. Theorems are built upon axioms:
Theorems are built upon axioms. They use axioms as the basis for their reasoning. Theorems cannot exist without axioms. They are dependent on the axioms for their validity.

4. Axioms are self-evident truths, while theorems are proven:
Axioms are statements that are accepted as true without proof. They are self-evident truths. Theorems, on the other hand, are proven to be true based on other statements that have been previously proven.

Conclusion

In conclusion, axiom and theorem are two terms that are closely related in mathematics. Axioms are the foundation of mathematics and theorems are built upon axioms. Theorems are derived from axioms and use axioms as the basis for their reasoning. Axioms are self-evident truths, while theorems are proven to be true based on other statements that have been previously proven.