Sure, let's solve this step-by-step to identify the best definitions for theorem, conjecture, and axiom.
1. Axiom:
- An axiom is a statement that is assumed to be true without proof. This means it is accepted as a foundational truth in mathematics and doesn't require a logical demonstration to be accepted.
2. Theorem:
- A theorem is a statement that has been shown to be true by rigorous application of logic. This involves proving the statement using a series of logical steps and previously established statements or axioms.
3. Conjecture:
- A conjecture is a statement that is believed to be true but hasn't been proven yet. It is an educated guess or hypothesis based on observation, but it requires a proof to be accepted as a theorem.
So the best definitions are:
- A statement that is assumed to be true without proof is an axiom.
- A statement that has been shown to be true by vigorous application of logic is a theorem.
- A statement that is believed to be true but hasn't been proven is a conjecture.
Let's fill these in the blanks properly:
- A statement that is assumed to be true without proof is an axiom.
- A statement that has been shown to be true by vigorous application of logic is a theorem.
- A statement that is believed to be true but hasn't been proven is a conjecture.
