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.
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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.
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.
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”).
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).
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.
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.