Overview
A corollary is a proposition that can be deduced easily from a previously established statement, such as a theorem or lemma. In ordinary usage it denotes a consequence that requires little additional proof beyond the result from which it follows. The term appears most often in formal reasoning, especially in mathematics, but it is also used in law, philosophy, and everyday language to mark an immediate implication of an earlier claim.
Characteristics and role
Typically a corollary has these features: it depends on a prior result, its justification is short or obvious once the earlier result is accepted, and it often clarifies or records useful consequences that would otherwise interrupt the flow of a long proof. Writers may label a consequence a corollary to emphasize that no substantial new ideas are required to establish it.
Relation to other terms
Corollaries, lemmas, propositions and theorems form a hierarchy of mathematical statements by role rather than by strict logical strength. A theorem is a major result, a lemma is an auxiliary result used in proving something else, a proposition is a result of intermediate importance, and a corollary is usually a quick consequence. This classification is partly conventional and depends on the author’s judgment about what deserves emphasis.
Examples and uses
Authors often present corollaries to record useful special cases or simple consequences. For example, after proving a general statement about continuous functions one might state corollaries about boundedness or intermediate values. In number theory, a theorem about prime divisors might immediately yield corollaries about divisibility patterns. Outside mathematics, people use the word to mean any natural consequence: for instance, "If taxes rise, a corollary may be reduced consumer spending."
Historical and practical notes
The word derives from Latin roots meaning "to flow together" and has been used in mathematical writing for centuries to structure arguments. Practically, listing corollaries improves readability: it separates main technical proofs from the convenient consequences that follow. When teaching or communicating results, clear corollaries help readers see how general theorems apply in specific contexts.
Common distinctions
- Immediate: a corollary typically needs minimal additional reasoning.
- Dependent: it presupposes the theorem from which it is drawn.
- Expository: used to highlight useful but secondary information.