Contour set

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

Formal definitions

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

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

Arithmetic

Consider a real number , and the relation . Then

Consider, more generally, the relation

Then

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

Economic

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

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

Complementarity

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

References

  1. 1 2 Robert P. Gilles (1996). Economic Exchange and Social Organization: The Edgeworthian Foundations of General Equilibrium Theory. Springer. p. 35.

Bibliography

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