Lemma (mathematics): Difference between revisions
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{{Distinguish|Lemma (morphology)}} | {{Distinguish|Lemma (morphology)}} | ||
In [[mathematics]] and other fields,{{efn|Such as [[informal logic]], [[argument mapping]], and [[philosophy]].<ref>[https://www.merriam-webster.com/dictionary/lemma.] "Lemma." Merriam-Webster.com Dictionary, Merriam-Webster.</ref><ref>Loewen, Nathan R. B. ''Beyond the Problem of Evil.'' Lexington Books. March 12, 2018. {{ISBN|9781498555739}} p. 47</ref>}} a '''lemma''' ({{plural form}}: '''lemmas''' or '''lemmata''') is a generally minor, proven [[Theorem#Terminology|proposition]] which is used to prove a larger statement. For that reason, it is also known as a "helping [[theorem]]" or an "auxiliary theorem".<ref>{{cite book |last= Higham |first= Nicholas J. |title= Handbook of Writing for the Mathematical Sciences |publisher= [[Society for Industrial and Applied Mathematics]] |year= 1998 |isbn= 0-89871-420-6 |pages= [https://archive.org/details/handbookofwritin0000high/page/16 16] |url= https://archive.org/details/handbookofwritin0000high/page/16 }}</ref><ref name=":0">{{Cite web|url=https://www.dictionary.com/browse/lemma|title=Definition of lemma {{!}} Dictionary.com|website=www.dictionary.com|language=en|access-date=2019-11-28}}</ref> In many cases, a lemma derives its importance from the theorem it aims to | In [[mathematics]] and other fields,{{efn|Such as [[informal logic]], [[argument mapping]], and [[philosophy]].<ref>[https://www.merriam-webster.com/dictionary/lemma.] "Lemma." Merriam-Webster.com Dictionary, Merriam-Webster.</ref><ref>Loewen, Nathan R. B. ''Beyond the Problem of Evil.'' Lexington Books. March 12, 2018. {{ISBN|9781498555739}} p. 47</ref>}} a '''lemma''' ({{plural form}}: '''lemmas''' or '''lemmata''') is a generally minor, [[mathematical proof|proven]] [[Theorem#Terminology|proposition]] which is used to prove a larger statement. For that reason, it is also known as a "helping [[theorem]]" or an "auxiliary theorem".<ref>{{cite book |last= Higham |first= Nicholas J. |title= Handbook of Writing for the Mathematical Sciences |publisher= [[Society for Industrial and Applied Mathematics]] |year= 1998 |isbn= 0-89871-420-6 |pages= [https://archive.org/details/handbookofwritin0000high/page/16 16] |url= https://archive.org/details/handbookofwritin0000high/page/16 }}</ref><ref name=":0">{{Cite web|url=https://www.dictionary.com/browse/lemma|title=Definition of lemma {{!}} Dictionary.com|website=www.dictionary.com|language=en|access-date=2019-11-28}}</ref> 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.<ref name=":1">{{Cite web|url=https://divisbyzero.com/2008/09/22/what-is-the-difference-between-a-theorem-a-lemma-and-a-corollary/|title=What is the difference between a theorem, a lemma, and a corollary?|last=Richeson|first=Dave|date=2008-09-23|website=David Richeson: Division by Zero|language=en|access-date=2019-11-28}}</ref> | ||
==Etymology== | ==Etymology== | ||
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* [[Lovász local lemma]] | * [[Lovász local lemma]] | ||
* [[Nakayama's lemma]] | * [[Nakayama's lemma]] | ||
* [[Noether normalization lemma]] | |||
* [[Poincaré lemma| Poincaré's lemma]] | * [[Poincaré lemma| Poincaré's lemma]] | ||
* [[Riesz's lemma]] | * [[Riesz's lemma]] | ||
| Line 34: | Line 35: | ||
* [[Vitali covering lemma]] | * [[Vitali covering lemma]] | ||
* [[Yoneda lemma|Yoneda's lemma]] | * [[Yoneda lemma|Yoneda's lemma]] | ||
* [[Zariski's lemma]] | |||
* [[Zorn's lemma]] | * [[Zorn's lemma]] | ||
{{div col end}} | {{div col end}} | ||
Latest revision as of 05:54, 23 October 2025
Template:Short description Script error: No such module "Distinguish".
In mathematics and other fields,Template:Efn a lemma (Template:Plural form: lemmas or lemmata) is a generally minor, proven proposition which is used to prove a larger statement. For that reason, it is also known as a "helping theorem" or an "auxiliary theorem".[1][2] 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.[3]
Etymology
From the Ancient Greek λῆμμα, (perfect passive εἴλημμαι) something received or taken. Thus something taken for granted in an argument.[4]
Comparison with theorem
There is no formal distinction between a lemma and a theorem, 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.[3]
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)
- Greendlinger's lemma
- Itô's lemma
- Jordan's lemma
- Lovász local lemma
- Nakayama's lemma
- Noether normalization 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
- Zariski'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.
See also
- Axiom
- Corollary
- Co-premise
- Fundamental lemma
- Inference objection
- List of lemmas
- Objection
- Porism
- Theorem
- Theorem terminology
Notes
References
External links
This article incorporates material from Lemma on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.