mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

and other fields, a lemma (: lemmas or lemmata) is a generally minor, proven

proposition A proposition is a statement that can be either true or false. It is a central concept in the philosophy of language, semantics, logic, and related fields. Propositions are the object s denoted by declarative sentences; for example, "The sky ...

which is used to prove a larger statement. For that reason, it is also known as a "helping

theorem In mathematics and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to esta ...

" or an "auxiliary theorem". In many cases, a lemma derives its importance from the theorem it aims to prove; however, a lemma can also turn out to be more important than originally thought.

Etymology

From the

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

λῆμμα, (perfect passive εἴλημμαι) something received or taken. Thus something taken for granted in an argument.

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

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 * Dehn's lemma * Euclid's lemma * Farkas' lemma * Fatou's lemma * Gauss's lemma (any of several named after

Carl Friedrich Gauss Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observatory and ...

) * Greendlinger's lemma * Itô's lemma * Jordan's lemma * Lovász local lemma * Nakayama's lemma * Poincaré's lemma * Riesz's lemma * Schur's lemma * Schwarz's lemma * Sperner's lemma * Urysohn's lemma * Vitali covering lemma * Yoneda's lemma *

Zorn's lemma Zorn's lemma, also known as the Kuratowski–Zorn lemma, is a proposition of set theory. It states that a partially ordered set containing upper bounds for every chain (that is, every totally ordered subset) necessarily contains at least on ...

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