There are four different equivalence relations which may be defined on the set of functions from to .

A preference relation, denoted , is a binary relation define on the set .

Many mathematical objects, such as sets of numbers and functions, exhibit internal structure as a consequence of operations or relations that are defined on the set.

Let be an open subset of the complex plane, a point of and a holomorphic function defined on the set .

The problem is that the difference is not defined on the set .

We can define an equivalence relation on the set of real numbers by means of this group of linear fractional transformations.

The Blum axioms can be used to define an abstract computational complexity theory on the set of computable functions.

So we have to define a suitable equivalence relation on the set of all parametric curves.

This reparametrisation of γ defines the equivalence relation on the set of all parametric C curves.

We need to show this integral coincides with the preceding one, defined on the set of simple functions.