In mathematics,

informal logic Informal logic encompasses the principles of logic and logical thought outside of a formal setting (characterized by the usage of particular statements). However, the precise definition of "informal logic" is a matter of some dispute. Ralph H. J ...

and argument mapping, a lemma (plural lemmas or lemmata) is a generally minor, proven

proposition In logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, "meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the no ...

which is used as a stepping stone to a larger result. For that reason, it is also known as a "helping

theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of ...

" or an "auxiliary theorem". In many cases, a lemma derives its importance from the theorem it aims to

prove Proof most often refers to: * Proof (truth), argument or sufficient evidence for the truth of a proposition * Alcohol proof, a measure of an alcoholic drink's strength Proof may also refer to: Mathematics and formal logic * Formal proof, a con ...

; however, a lemma can also turn out to be more important than originally thought. The word "lemma" derives from the

Ancient Greek Ancient Greek includes the forms of the Greek language used in ancient Greece and the ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek (), Dark Ages (), the Archaic p ...

("anything which is received", such as a gift, profit, or a bribe).

Comparison with theorem

There is no formal distinction between a lemma and a

, only one of intention (see Theorem terminology). However, a lemma can be considered a minor result whose sole purpose is to help prove a more substantial theorem – a step in the direction of proof.

Well-known lemmas

A good stepping stone can lead to many others. Some powerful results in mathematics are known as lemmas, first named for their originally minor purpose. These include, among others: * Bézout's lemma *

Burnside's lemma Burnside's lemma, sometimes also called Burnside's counting theorem, the Cauchy–Frobenius lemma, the orbit-counting theorem, or the Lemma that is not Burnside's, is a result in group theory that is often useful in taking account of symmetry when ...

Dehn's lemma In mathematics, Dehn's lemma asserts that a piecewise-linear map of a disk into a 3-manifold, with the map's singularity set in the disk's interior, implies the existence of another piecewise-linear map of the disk which is an embedding and is id ...

* Euclid's lemma *

Farkas' lemma Farkas' lemma is a solvability theorem for a finite system of linear inequalities in mathematics. It was originally proven by the Hungarian mathematician Gyula Farkas. Farkas' lemma is the key result underpinning the linear programming duality an ...

* Fatou's lemma *

Gauss's lemma Gauss's lemma can mean any of several lemmas named after Carl Friedrich Gauss: * * * * A generalization of Euclid's lemma is sometimes called Gauss's lemma See also * List of topics named after Carl Friedrich Gauss Carl Friedrich Gauss ( ...

* Greendlinger's lemma * Itô's lemma * Jordan's lemma *

Nakayama's lemma In mathematics, more specifically abstract algebra and commutative algebra, Nakayama's lemma — also known as the Krull–Azumaya theorem — governs the interaction between the Jacobson radical of a ring (typically a commutative ring) a ...

* Poincaré's lemma * Riesz's lemma * Schur's lemma *

Schwarz's lemma In mathematics, the Schwarz lemma, named after Hermann Amandus Schwarz, is a result in complex analysis about holomorphic functions from the open unit disk to itself. The lemma is less celebrated than deeper theorems, such as the Riemann mapping ...

* Sperner's lemma * Urysohn's lemma *

Vitali covering lemma In mathematics, the Vitali covering lemma is a combinatorial and geometric result commonly used in measure theory of Euclidean spaces. This lemma is an intermediate step, of independent interest, in the proof of the Vitali covering theorem. The co ...

* Yoneda's lemma * Zorn's lemma While these results originally seemed too simple or too technical to warrant independent interest, they have eventually turned out to be central to the theories in which they occur.

Notes

References

External links

Doron Zeilberger Doron Zeilberger (דורון ציילברגר, born 2 July 1950 in Haifa, Israel) is an Israeli mathematician, known for his work in combinatorics. Education and career He received his doctorate from the Weizmann Institute of Science in 1976, ...

Opinion 82: A Good Lemma is Worth a Thousand Theorems
{{PlanetMath attribution, id=4492, title=Lemma Mathematical terminology *