Toy theorem
Encyclopedia
In mathematics
, a toy theorem is a simplified version of a more general theorem
. For instance, by introducing some simplifying assumptions in a theorem, one obtains a toy theorem.
Usually, a toy theorem is used to illustrate the claim of a theorem. It can also be insightful to study proofs of a toy theorem derived from a non-trivial theorem. Toy theorems can also have education value. 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.
For instance, 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
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a toy theorem is a simplified version of a more general theorem
Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...
. For instance, by introducing some simplifying assumptions in a theorem, one obtains a toy theorem.
Usually, a toy theorem is used to illustrate the claim of a theorem. It can also be insightful to study proofs of a toy theorem derived from a non-trivial theorem. Toy theorems can also have education value. 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.
For instance, a toy theorem of the Brouwer fixed point theorem
Brouwer fixed point theorem
Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after Luitzen Brouwer. It states that for any continuous function f with certain properties there is a point x0 such that f = x0. The simplest form of Brouwer's theorem is for continuous functions f from a disk D to...
is obtained by restricting the dimension
Dimension
In physics and mathematics, the dimension of a 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 because only one coordinate is needed to specify a point on it...
to one. In this case, the Brouwer fixed point theorem follows almost immediately from the intermediate value theorem
Intermediate value theorem
In mathematical analysis, the intermediate value theorem states that for each value between the least upper bound and greatest lower bound of the image of a continuous function there is at least one point in its domain that the function maps to that value....
.