In mathematics, contour sets generalize and formalize the everyday notions of

  • everything superior to something
  • everything superior or equivalent to something
  • everything inferior to something
  • everything inferior or equivalent to something.

Formal definitions

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Given a relation on pairs of elements of set  

 

and an element   of  

 

The upper contour set of   is the set of all   that are related to  :

 

The lower contour set of   is the set of all   such that   is related to them:

 

The strict upper contour set of   is the set of all   that are related to   without   being in this way related to any of them:

 

The strict lower contour set of   is the set of all   such that   is related to them without any of them being in this way related to  :

 

The formal expressions of the last two may be simplified if we have defined

 

so that   is related to   but   is not related to  , in which case the strict upper contour set of   is

 

and the strict lower contour set of   is

 

Contour sets of a function

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In the case of a function   considered in terms of relation  , reference to the contour sets of the function is implicitly to the contour sets of the implied relation

 

Examples

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Arithmetic

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Consider a real number  , and the relation  . Then

  • the upper contour set of   would be the set of numbers that were greater than or equal to  ,
  • the strict upper contour set of   would be the set of numbers that were greater than  ,
  • the lower contour set of   would be the set of numbers that were less than or equal to  , and
  • the strict lower contour set of   would be the set of numbers that were less than  .

Consider, more generally, the relation

 

Then

  • the upper contour set of   would be the set of all   such that  ,
  • the strict upper contour set of   would be the set of all   such that  ,
  • the lower contour set of   would be the set of all   such that  , and
  • the strict lower contour set of   would be the set of all   such that  .

It would be technically possible to define contour sets in terms of the relation

 

though such definitions would tend to confound ready understanding.

In the case of a real-valued function   (whose arguments might or might not be themselves real numbers), reference to the contour sets of the function is implicitly to the contour sets of the relation

 

Note that the arguments to   might be vectors, and that the notation used might instead be

 

Economics

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In economics, the set   could be interpreted as a set of goods and services or of possible outcomes, the relation   as strict preference, and the relationship   as weak preference. Then

  • the upper contour set, or better set,[1] of   would be the set of all goods, services, or outcomes that were at least as desired as  ,
  • the strict upper contour set of   would be the set of all goods, services, or outcomes that were more desired than  ,
  • the lower contour set, or worse set,[1] of   would be the set of all goods, services, or outcomes that were no more desired than  , and
  • the strict lower contour set of   would be the set of all goods, services, or outcomes that were less desired than  .

Such preferences might be captured by a utility function  , in which case

  • the upper contour set of   would be the set of all   such that  ,
  • the strict upper contour set of   would be the set of all   such that  ,
  • the lower contour set of   would be the set of all   such that  , and
  • the strict lower contour set of   would be the set of all   such that  .

Complementarity

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On the assumption that   is a total ordering of  , the complement of the upper contour set is the strict lower contour set.

 
 

and the complement of the strict upper contour set is the lower contour set.

 
 

See also

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References

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  1. ^ a b Robert P. Gilles (1996). Economic Exchange and Social Organization: The Edgeworthian Foundations of General Equilibrium Theory. Springer. p. 35. ISBN 9780792342007.

Bibliography

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