What's the difference between theorem, lemma and corollary?

Question

Can anybody explain me what is the basic difference between theorem, lemma and corollary?

We have been using it for a long time but I never paid any attention. I am just curious to know.

Answer 1

A lot of authors like to use lemma to mean "small theorem." Often a group of lemmas are used to prove a larger result, a "theorem."

A corollary is something that follows trivially from any one of a theorem, lemma, or other corollary.

However, when it boils down to it, all of these things are equivalent as they denote the truth of a statement.

Answer 2

Lemma is generally used to describe a "helper" fact that is used in the proof of a more significant result.

Significant results are frequently called theorems.

Short, easy results of theorems are called corollaries.

But the words aren't exactly that set in stone.

Answer 3

A lemma are those minor results which are used into proving a definite results of a theorem.

Answer 4

lemma: A basic result which are used to prove theorems
theorem :Relatively more important and big result which has to be proved corollary: special case result which intuitively comes from theorem. conjecture:A result which is assumed to be true but still not prove exists. Proposition: A result which is either true or false. Axioms or postulates: A set of statement without proof which is assumed to be true and used building blocks to prove several mathematical theorems and results.

Answer 5

My 2 cents on the terminology:

A theorem is a proven statement.

Both lemma and corollary are (special kinds of) theorems.

The "usual" difference is that a lemma is a minor theorem usually towards proving a more significant theorem. Whereas a corollary is an "easy" or "evident" consequence of another theorem (or lemma).

Axiom or postulate is a statement that is taken as true without proof (usually a self-evident or known to hold truth or simply assumed true).

A definition, according to a known mathematician, is a special kind of postulate, which introduces (new) concepts later used in other propositions.

Answer 6

John Playfair, in his 1862 "Elements of Geometry" (textbook) gives a numbered list as follows (I quote): (5) A Theorem is a demonstrative proposition; in which some property is asserted, and the truth of it is required to be proved. ... (10) A Lemma is a preparatory proposition, laid down in order to shorten the demonstration of the main proposition which follows it. (12) A Corollary, ..., is a consequence drawn immediately from some proposition or other premises.

Answer 7

1. A definition is an identity to mathematical property.

For example, we can define parallel lines as coplanar straight lines that never meet however they are produced. Similary, we can say that "an angle in a semicircle is that which is subtended by a diameter", that is a definition. However, if we say "An angle in a semicircle is a right angle" that is not a definition.

2. An axiom is a property that is accepted to be true but has no proof.

For example, the stetement "if two straight lines are parallel and are crossed by transversal, the corresponding angles so formed are equal" is an axiom. It's hard to prove it by mathematical definitions but it is true.

3. A lemma is simply a property that can be proved by definitions and axioms only.

For example, the statement "If two parallel straight lines are joined by a transversal, the interior angles so formed are supplementary" is a lemma. The axiom of corresponding angles, and that of adjacent angles on the straight line, can be used to prove this property and hence its a lemma.

4. A theorem is a property that can be proved by at least one lemma.

For example, the statement "opposite angles of a parallelogram are equal" is a theorem because you can prove it by a lemma of interior angles between parallel lines.

In many mathematical books are statements that are wrongly called lemmas when they are in fact theorems because the writer thinks they are small. In pure mathematics papers, a lemma can be taken as a theorem from another paper, which I think is dishonest.