In
mathematics, a toy theorem is a simplified instance (
special case
In logic, especially as applied in mathematics, concept is a special case or specialization of concept precisely if every instance of is also an instance of but not vice versa, or equivalently, if is a generalization of . A limiting case i ...
) of a more general
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 t ...
, which can be useful in providing a handy representation of the general theorem, or a framework for proving the general theorem. One way of obtaining a toy theorem is by introducing some simplifying assumptions in a theorem.
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.
Examples
A toy theorem of the
Brouwer fixed-point theorem
Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. (Bertus) Brouwer. It states that for any continuous function f mapping a compact convex set to itself there is a point x_0 such that f(x_0)=x_0. The simples ...
is obtained by restricting the
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
to one. In this case, the Brouwer fixed-point theorem follows almost immediately from the
intermediate value theorem.
Another example of toy theorem is
Rolle's theorem, which is obtained from the
mean value theorem by equating the
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
values at the endpoints.
See also
*
Corollary
*
Fundamental theorem
*
Lemma (mathematics)
In mathematics, informal logic and argument mapping, a lemma (plural lemmas or lemmata) is a generally minor, proven proposition which is used as a stepping stone to a larger result. For that reason, it is also known as a "helping theorem" or an ...
*
Toy model
In the modeling of physics, a toy model is a deliberately simplistic model with many details removed so that it can be used to explain a mechanism concisely. It is also useful in a description of the fuller model.
* In "toy" mathematical models ...
References
{{PlanetMath attribution, id=4684, title=toy theorem
Mathematical theorems
Mathematical terminology