↑ arvau

Wikipedia / Contour set

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

Given a relation on pairs of elementss of set

The upper contour set:

The lower contour set is related to them:

The strict upper contour set being in this way related to any of them:

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

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

is not is

is

Contour sets of a function

In the case of a function, 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 \ge. Then

  • would be the set of numbers that were greater than or equal,

  • the strict would be the set of numbers that were greater,

  • would be the set of numbers that were less than or equal, and

  • the strict would be the set of numbers that were less.

Consider, more generally, the relation

Then

  • ,

  • the strict,

  • , and

  • the strict.

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

though such definitions would tend to confound ready understanding.

(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

might be vectorss, and that the notation used might instead be

Economics

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

  • the upper contour set, or better set, would be the set of all goods, services, or outcomes that were at least as desired,

  • the strict would be the set of all goods, services, or outcomes that were more desired,

  • the lower contour set, or worse set, would be the set of all goods, services, or outcomes that were no more desired, and

  • the strict would be the set of all goods, services, or outcomes that were less desired.

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

  • ,

  • the strict,

  • , and

  • the strict.

Complementarity

is a total ordering, 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

Bibliography

  • Andreu Mas-Colell, Michael D. Whinston, and Jerry R. Green, Microeconomic Theory (), p43. (cloth) (paper)