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.
Complex numbers for an example, and the idea of an equivalence relation to set things in context.
Example 1: Möbius transformations and triples of points
A Möbius map, or Möbius transformation, is a function from to given by a formula of the form , where , , and are complex numbers with . Such maps form a group under composition, as discussed in an example in the article Using generators and closure properties. We will consider Möbius transformations acting on complex numbers, as well as on the special symbol ; indeed, they act on the Riemann sphere
Consider the collection of triples of distinct points in It turns out that any such triple of points can be mapped to any other triple of distinct points by a Möbius transformation. If one attempts to show this directly by finding suitable values for by substitution, however, the algebra quickly gets quite messy. This is where the technique of this article comes in: if the triple is supposed to be mappable to any other triple , then one must certainly be able to map it to the simple triple . But if one can do that for any triple , then to show that can be mapped to in general one can simply take the composition of a map that takes to with the inverse of one that takes to Thus once one has established that all triples are 'equivalent' to the simple case , one is done. Furthermore, establishing this last fact is relatively simple, by virtue of having chosen a simple representative.
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 transitive relation, meaning that if X is related to Y and Y is related to Z, then X is related to Z.♦