Uncovering the Importance of Lemmas in Mathematics

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In summary, a lemma is a statement used to prove something else, typically considered less interesting than a theorem. These are often used in mathematics, such as in Zorn's lemma, to aid in proving larger concepts. "Corollaries" are statements that follow easily from a theorem, while "lemmas" are typically used to prove a theorem.
  • #1
Char. Limit
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What is a lemma?

I hear people talking about "lemmas" all the time. A great example is the math joke that ends with the devil saying "But I found this really interesting lemma..." This joke would likely be much funnier to me if I knew what a lemma (or Riemann's hypothesis, but if you want to, I don't necessarily need to know that) is.

All help appreciated.
 
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  • #2


A lemma is a statement you can prove so that you can prove something else. The only reason we don't call them theorems is that they are typically aren't interesting enough to "deserve" that title.
 
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  • #3


You might also want to note "corrollary". A corrollary is a statement that is not important enough be called a "theorem" in its own right but follows easily from a given theorem.

"Lemmas" are used to prove a theorem so are proved before the theorem. "Corrollaries" are proved from the theorem an so are proved after the theorem.
 
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  • #4


Thanks for the help.
 
  • #5


There are SOME lemmas, though, that have shown themselves to be far more useful than the theorem originally sought to be proven by the aid of that lemma..

Zorn's lemma, for example.
 

What is a Lemma?

A lemma is a term used in linguistics and mathematics to refer to a unit of language or a basic concept that is used as a building block for more complex words or ideas. It is often considered the smallest meaningful unit of a language or a logical statement.

How is a Lemma different from a word?

A lemma differs from a word in that it is an abstract concept, while a word is a concrete instance of that concept. For example, the lemma "run" encompasses all forms of the verb, such as "running" and "ran". A word, on the other hand, refers to a specific form of the verb, such as "ran".

What is the purpose of using Lemmas?

Lemmas are used in linguistic and mathematical analysis to break down complex language or logical statements into their basic components. This allows for easier comparison and analysis of different forms of a word or concept.

How are Lemmas identified?

In linguistics, Lemmas are identified through a process called lemmatization, which involves grouping different forms of a word together under a single lemma. In mathematics, Lemmas are often assigned numbers or symbols to represent them in logical statements.

Do all languages use Lemmas?

No, not all languages use Lemmas in their linguistic or mathematical analysis. Some languages, such as Chinese, do not have a concept of lemmas and instead rely on individual characters or words to convey meaning.

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