In mathematics, a toy theorem is a simplified instance (special case) of a more general theorem, which can be useful in providing a handy representation of the general theorem, or a framework for proving the general theorem.[1] One way of obtaining a toy theorem is by introducing some simplifying assumptions in a theorem.[2]
In many cases, a toy theorem is used to illustrate the claim of a theorem, while in other cases, studying the proofs of a toy theorem (derived from a non-trivial theorem) can provide insight that would be hard to obtain otherwise.
Toy theorems can also have educational value as well. For example, after presenting a theorem (with, say, a highly non-trivial proof), one can sometimes give some assurance that the theorem really holds, by proving a toy version of the theorem.[2]
Examples
A toy theorem of the Brouwer fixed-point theorem is obtained by restricting the dimension to one. In this case, the Brouwer fixed-point theorem follows almost immediately from the intermediate value theorem.[2]
Another example of toy theorem is Rolle's theorem, which is obtained from the mean value theorem by equating the function value at the endpoints.[1]
See also
References
- ^ a b "The Definitive Glossary of Higher Mathematical Jargon — Theorem". Math Vault. 2019-08-01. Retrieved 2019-11-26.
- ^ a b c "toy theorem". planetmath.org. Retrieved 2019-11-26.