Final answer:
The correct options are 1) axiom, theorem for a statement that is assumed true without proof and one that is proven true, and 3) axiom, hypothesis for a statement that is believed true but not proven.
Step-by-step explanation:
A statement that is assumed to be true without proof is a(n) axiom of logic. A statement that has been shown to be true by rigorous application and reasoning is a theorem. A statement that is believed to be true but hasn't been proven is a hypothesis. Therefore, the correct options are 1) axiom, theorem and 3) axiom, hypothesis.
In the context of logic and mathematics, an axiom is a foundational statement or principle that is accepted without proof. These axioms form the basis upon which theorems are derived. When an axiom or set of axioms is applied to deduce a statement rigorously, and that statement is proven, it becomes a theorem. Logical arguments and proofs consist of premises — these are axioms or hypotheses — that lead to conclusions through logical reasoning. Unlike a hypothesis, which is a tentative statement that explains an observation and requires testing, an axiom is taken as universally true within its system without requiring empirical verification.