Editing Weak ordering (section)

===Representation by functions===

For sets of sufficiently small [[cardinality]], a fourth axiomatization is possible, based on real-valued functions. If <math>X</math> is any set then a real-valued function <math>f : X \to \R</math> on <math>X</math> induces a strict weak order on <math>X</math> by setting 
<math display=block>a < b \text{ if and only if } f(a) < f(b).</math> 
The associated total preorder is given by setting 
<math>a {}\lesssim{} b \text{ if and only if } f(a) \leq f(b)</math> 
and the associated equivalence by setting 
<math>a {}\sim{} b \text{ if and only if } f(a) = f(b).</math>

The relations do not change when <math>f</math> is replaced by <math>g \circ f</math> ([[Function composition|composite function]]), where <math>g</math> is a [[Monotonic function|strictly increasing]] real-valued function defined on at least the range of <math>f.</math> Thus for example, a [[Utility#Utility functions|utility function]] defines a [[preference]] relation. In this context, weak orderings are also known as '''preferential arrangements'''.<ref>{{citation|last=Gross|first=O. A.|doi=10.2307/2312725|journal=The American Mathematical Monthly|mr=0130837|pages=4–8|title=Preferential arrangements|volume=69|year=1962|issue=1|jstor=2312725}}.</ref>

If <math>X</math> is finite or countable, every weak order on <math>X</math> can be represented by a function in this way.<ref name="roberts">{{citation|last=Roberts|first=Fred S.|authorlink=Fred S. Roberts|year=1979|title=Measurement Theory, with Applications to Decisionmaking, Utility, and the Social Sciences|at=[https://archive.org/details/measurementtheor0000robe/page/ Theorem 3.1]|series=Encyclopedia of Mathematics and its Applications|volume=7|isbn=978-0-201-13506-0|publisher=Addison-Wesley|url-access=registration|url=https://archive.org/details/measurementtheor0000robe/page/}}.</ref> However, there exist strict weak orders that have no corresponding real function. For example, there is no such function for the [[lexicographic order]] on <math>\R^n.</math> Thus, while in most preference relation models the relation defines a utility function [[up to]] order-preserving transformations, there is no such function for [[lexicographic preferences]].

More generally, if <math>X</math> is a set, <math>Y</math> is a set with a strict weak ordering <math>\,<,\,</math> and <math>f : X \to Y</math> is a function, then <math>f</math> induces a strict weak ordering on <math>X</math> by setting 
<math display=block>a < b \text{ if and only if } f(a) < f(b).</math>
As before, the associated total preorder is given by setting 
<math>a {}\lesssim{} b \text{ if and only if } f(a) {}\lesssim{} f(b),</math> 
and the associated equivalence by setting 
<math>a {}\sim{} b \text{ if and only if } f(a) {}\sim{} f(b).</math> 
It is not assumed here that <math>f</math> is an [[injective function]], so a class of two equivalent elements on <math>Y</math> may induce a larger class of equivalent elements on <math>X.</math> Also, <math>f</math> is not assumed to be a [[surjective function]], so a class of equivalent elements on <math>Y</math> may induce a smaller or empty class on <math>X.</math> However, the function <math>f</math> induces an injective function that maps the partition on <math>X</math> to that on <math>Y.</math> Thus, in the case of finite partitions, the number of classes in <math>X</math> is less than or equal to the number of classes on <math>Y.</math>