111
$\begingroup$

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.

$\endgroup$
7
  • 18
    $\begingroup$ Lemmas are smaller results to be used in a bigger (more important) result. The big result is usually a theorem. Corollaries are special cases of theorems. $\endgroup$ Aug 9, 2013 at 5:23
  • $\begingroup$ @CameronWilliams... Oh.... thanks a lot. Your reply in few words cleared my doubt :) $\endgroup$
    – monalisa
    Aug 9, 2013 at 5:24
  • 1
    $\begingroup$ I'm glad I could help :) $\endgroup$ Aug 9, 2013 at 5:25
  • 4
    $\begingroup$ @Cameron is, of course, correct but let me add that lemmas needn't always be small results. Typically, they are but some results such as Urysohn's lemma could also be referred to as theorems. See, e.g., math.stackexchange.com/questions/111428/lemma-vs-theorem for additional discussion. $\endgroup$ Aug 9, 2013 at 5:33
  • 2
    $\begingroup$ See also en.wikipedia.org/wiki/List_of_lemmas for a list of famous lemmas. $\endgroup$
    – lhf
    Nov 25, 2014 at 11:31

10 Answers 10

81
$\begingroup$

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.

$\endgroup$
1
  • 1
    $\begingroup$ so it seems like, the one we don't really need to write down the proof is corollary?? $\endgroup$ Sep 22, 2018 at 5:53
57
$\begingroup$

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.

$\endgroup$
1
  • 24
    $\begingroup$ I want to stress that calling something a lemma, theorem, or corollary is purely a choice made for organizational purposes. That's probably the most important thing to take away from my answer: a lemma is just as true as a theorem! $\endgroup$
    – parsiad
    Aug 9, 2013 at 5:27
40
$\begingroup$

Terence Tao (Analysis I, p. 25, n. 4):

From a logical point of view, there is no difference between a lemma, proposition, theorem, or corollary - they are all claims waiting to be proved. However, we use these terms to suggest different levels of importance and difficulty.

A lemma is an easily proved claim which is helpful for proving other propositions and theorems, but is usually not particularly interesting in its own right.

A proposition is a statement which is interesting in its own right, while

a theorem is a more important statement than a proposition which says something definitive on the subject, and often takes more effort to prove than a proposition or lemma.

A corollary is a quick consequence of a proposition or theorem that was proven recently.

$\endgroup$
18
$\begingroup$

Here is some information from this link:

Theorem — a mathematical statement that is proved using rigorous mathematical reasoning. In a mathematical paper, the term theorem is often reserved for the most important results.

Lemma — a minor result whose sole purpose is to help in proving a theorem. It is a stepping stone on the path to proving a theorem. Very occasionally lemmas can take on a life of their own (Zorn’s lemma, Urysohn’s lemma, Burnside’s lemma, Sperner’s lemma).

Corollary — a result in which the (usually short) proof relies heavily on a given theorem (we often say that “this is a corollary of Theorem A”).

$\endgroup$
3
  • $\begingroup$ So a result which relies heavily on a lemma or proposition (but not a theorem) is not a corollary? $\endgroup$
    – ashpool
    Aug 5, 2015 at 5:32
  • 12
    $\begingroup$ Copying text verbatim is not a helpful answer. Especially when the source is not mentioned/linked (divisbyzero.com/2008/09/22/…) $\endgroup$
    – Neowizard
    Aug 2, 2016 at 16:57
  • 1
    $\begingroup$ It's still quite helpful $\endgroup$
    – voices
    Aug 20, 2019 at 11:48
5
$\begingroup$

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

$\endgroup$
5
$\begingroup$

When writing a paper, i usually call lemma a technical result that will be used many times in the remaining part of the paper. The idea is to avoid repeating a similar argument in different proofs. For that reason a lemma is not always worth remembering in itself (and if it is, one can include it in a broader theorem).

$\endgroup$
0
5
$\begingroup$

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.

$\endgroup$
3
$\begingroup$

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.

$\endgroup$
0
$\begingroup$
  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.

  1. 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.

  1. 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.

  1. 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.

$\endgroup$
-2
$\begingroup$

parsiad (above) describes Theorems, Corollaries Lemmas as TRUTHS. Although that's true, it's misleading. These 3 are all Theorems, in the broader sense of the term, because ultimately Truth is the aim of an axiomatic method in mathematics. However, the truth is Conditional on the Truth of the Axioms, which are merely hypothesized to be so; any axiomatic system, can be changed by replacing an independent axiom with it's Negation.

The essential condition of all three is that they be Provable (not True), which means that they follow from the Axioms, or other Theorems, by mere Logical Deduction.

That said, Corollaries and Lemmas are Theorems judged within Metamathematics, so to speak, as to how Trivial or Relevant they are to the ultimate Proofs one seeks. One of these is merely an intermediary Theorem for another Theorem, and the other is a an easy result from a previous Theorem (or Axiom).

$\endgroup$

Not the answer you're looking for? Browse other questions tagged or ask your own question.