Make use of special cases and transitivity

a repository of mathematical know-how

Make use of special cases and transitivity

Title: *

Area of mathematics: *

A comma-separated list of areas of mathematics to which this article applies. Use ">" to tag in a subcategory. Example: Analysis > Harmonic analysis, Combinatorics

Keywords:

A comma-separated list of keywords associated with this article. Example: free group

Used in:

A comma-separated list of examples of where this technique is used. Example: Cauchy-Schwarz inequality

Parent articles:		Order

Body:

[QUICK DESCRIPTION]
To establish a result for all objects X, it sometimes suffices to establish it for some relatively simple special cases from which the full result can more straightforwardly be deduced. A typical situation might be that one wants to show that two collections of objects X and Y are 'equivalent'; the easiest way to do so might be to show that they are both 'equivalent' to a third, simple collection of objects Z.

[note article contributions wanted]More examples needed. [/note]

[PREREQUISITES]
Complex numbers for an example, and the idea of an [[w:Equivalence_relation|equivalence relation]] to set things in context.

[EXAMPLE mobius|M\"obius transformations and triples of points]
A ''M\"obius map'', or ''[[w:Möbius_transformation|M\"obius transformation]]'', is a function from $\C$ to $\C$ given by a formula of the form $z\mapsto\frac{az+b}{cz+d}$, where $a$, $b$, $c$ and $d$ are complex numbers with $ad-bc\ne 0$. Such maps form a group under composition, as discussed in an example in the article [[Using generators and closure properties]]. We will consider M\"obius transformations acting on complex numbers, as well as on the special symbol $\infty$; indeed, they act on the [[w:Riemann_sphere|Riemann sphere]] $\C \cup \{ \infty \}.$

Consider the collection of triples $(x,y,z)$ of distinct points in $\C \cup \{ \infty \}.$ It turns out that any such triple of points can be mapped to any other triple $(x',y',z')$ of distinct points by a M\"obius transformation. If one attempts to show this directly by finding suitable values for $a,b,c,d$ by substitution, however, the algebra quickly gets quite messy. This is where the technique of this article comes in: if the triple $(x,y,z)$ is supposed to be mappable to any other triple $(x',y',z')$, then one must certainly be able to map it to the simple triple $(0,1,\infty)$. But if one can do that for any triple $(x,y,z)$, then to show that $(x,y,z)$ can be mapped to $(x',y',z')$ in general one can simply take the composition of a map that takes $(x,y,z)$ to $(0,1,\infty)$ with the inverse of one that takes $(x',y',z')$ to $(0,1,\infty).$ Thus once one has established that all triples are 'equivalent' to the simple case $(0,1,\infty)$, one is done. Furthermore, establishing this last fact is relatively simple, by virtue of having chosen a simple representative.

[GENERAL DISCUSSION]
In the above example, there was a natural choice for the 'simple special case' for which it sufficed to establish the result. A choice of such an object may not always be so obvious, but can often be found after a little thought. The technique can also be applied in situations where the notion of 'equivalence' is slightly more rough, not actually being an equivalence relation but just a [add]transitive relation{{, meaning that if X is related to Y and Y is related to Z, then X is related to Z}}[/add].<comment thread="15" />

This is a stub

A stub is an article that is not sufficiently complete to be interesting.

Notifications

File attachments

Changes made to the attachments are not permanent until you save this post. The first "listed" file will be included in RSS feeds.

Attach new file:

Images are larger than 640x480 will be resized. The maximum upload size is 1 MB. Only files with the following extensions may be uploaded: jpg jpeg gif png svg.

Revision information

Log message:

An explanation of the additions or updates being made to help other authors understand your motivations.

Make use of special cases and transitivity

Recent articles

Active forum topics

Recent comments