A theorem is a statement or proposition that has been proven to be true based on a set of axioms and previously established statements. It is often expressed in clear mathematical terms, demonstrating a certain relationship or property that holds under specific conditions.
A proof, on the other hand, is a logical argument that validates the truth of a theorem. It consists of a series of statements and deductions that lead from accepted axioms and previously proven theorems to the conclusion that the theorem holds true.
In simple terms, you can think of a theorem as the ‘claim’ and the proof as the ‘justification’ for that claim. Theorem without proof remains unverified; proof without a theorem lacks context. This relationship is crucial in the discipline of mathematics and logic, where proving theorems builds the foundation of mathematical knowledge.