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 elements of set [math]\displaystyle{ X }[/math]

[math]\displaystyle{ \succcurlyeq~\subseteq~X^2 }[/math]

and an element [math]\displaystyle{ x }[/math] of [math]\displaystyle{ X }[/math]

[math]\displaystyle{ x\in X }[/math]

The upper contour set of [math]\displaystyle{ x }[/math] is the set of all [math]\displaystyle{ y }[/math] that are related to [math]\displaystyle{ x }[/math]:

[math]\displaystyle{ \left\{ y~\backepsilon~y\succcurlyeq x\right\} }[/math]

The lower contour set of [math]\displaystyle{ x }[/math] is the set of all [math]\displaystyle{ y }[/math] such that [math]\displaystyle{ x }[/math] is related to them:

[math]\displaystyle{ \left\{ y~\backepsilon~x\succcurlyeq y\right\} }[/math]

The strict upper contour set of [math]\displaystyle{ x }[/math] is the set of all [math]\displaystyle{ y }[/math] that are related to [math]\displaystyle{ x }[/math] without [math]\displaystyle{ x }[/math] being in this way related to any of them:

[math]\displaystyle{ \left\{ y~\backepsilon~(y\succcurlyeq x)\land\lnot(x\succcurlyeq y)\right\} }[/math]

The strict lower contour set of [math]\displaystyle{ x }[/math] is the set of all [math]\displaystyle{ y }[/math] such that [math]\displaystyle{ x }[/math] is related to them without any of them being in this way related to [math]\displaystyle{ x }[/math]:

[math]\displaystyle{ \left\{ y~\backepsilon~(x\succcurlyeq y)\land\lnot(y\succcurlyeq x)\right\} }[/math]

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

[math]\displaystyle{ \succ~=~\left\{ \left(a,b\right)~\backepsilon~\left(a\succcurlyeq b\right)\land\lnot(b\succcurlyeq a)\right\} }[/math]

so that [math]\displaystyle{ a }[/math] is related to [math]\displaystyle{ b }[/math] but [math]\displaystyle{ b }[/math] is not related to [math]\displaystyle{ a }[/math], in which case the strict upper contour set of [math]\displaystyle{ x }[/math] is

[math]\displaystyle{ \left\{ y~\backepsilon~y\succ x\right\} }[/math]

and the strict lower contour set of [math]\displaystyle{ x }[/math] is

[math]\displaystyle{ \left\{ y~\backepsilon~x\succ y\right\} }[/math]

Contour sets of a function

In the case of a function [math]\displaystyle{ f() }[/math] considered in terms of relation [math]\displaystyle{ \triangleright }[/math], reference to the contour sets of the function is implicitly to the contour sets of the implied relation

[math]\displaystyle{ (a\succcurlyeq b)~\Leftarrow~[f(a)\triangleright f(b)] }[/math]

Examples

Arithmetic

Consider a real number [math]\displaystyle{ x }[/math], and the relation [math]\displaystyle{ \ge }[/math]. Then

• the upper contour set of [math]\displaystyle{ x }[/math] would be the set of numbers that were greater than or equal to [math]\displaystyle{ x }[/math],
• the strict upper contour set of [math]\displaystyle{ x }[/math] would be the set of numbers that were greater than [math]\displaystyle{ x }[/math],
• the lower contour set of [math]\displaystyle{ x }[/math] would be the set of numbers that were less than or equal to [math]\displaystyle{ x }[/math], and
• the strict lower contour set of [math]\displaystyle{ x }[/math] would be the set of numbers that were less than [math]\displaystyle{ x }[/math].

Consider, more generally, the relation

[math]\displaystyle{ (a\succcurlyeq b)~\Leftarrow~[f(a)\ge f(b)] }[/math]

Then

• the upper contour set of [math]\displaystyle{ x }[/math] would be the set of all [math]\displaystyle{ y }[/math] such that [math]\displaystyle{ f(y)\ge f(x) }[/math],
• the strict upper contour set of [math]\displaystyle{ x }[/math] would be the set of all [math]\displaystyle{ y }[/math] such that [math]\displaystyle{ f(y)\gt f(x) }[/math],
• the lower contour set of [math]\displaystyle{ x }[/math] would be the set of all [math]\displaystyle{ y }[/math] such that [math]\displaystyle{ f(x)\ge f(y) }[/math], and
• the strict lower contour set of [math]\displaystyle{ x }[/math] would be the set of all [math]\displaystyle{ y }[/math] such that [math]\displaystyle{ f(x)\gt f(y) }[/math].

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

[math]\displaystyle{ (a\succcurlyeq b)~\Leftarrow~[f(a)\le f(b)] }[/math]

though such definitions would tend to confound ready understanding.

In the case of a real-valued function [math]\displaystyle{ f() }[/math] (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

[math]\displaystyle{ (a\succcurlyeq b)~\Leftarrow~[f(a)\ge f(b)] }[/math]

Note that the arguments to [math]\displaystyle{ f() }[/math] might be vectors, and that the notation used might instead be

[math]\displaystyle{ [(a_1 ,a_2 ,\ldots)\succcurlyeq(b_1 ,b_2 ,\ldots)]~\Leftarrow~[f(a_1 ,a_2 ,\ldots)\ge f(b_1 ,b_2 ,\ldots)] }[/math]

Economics

In economics, the set [math]\displaystyle{ X }[/math] could be interpreted as a set of goods and services or of possible outcomes, the relation [math]\displaystyle{ \succ }[/math] as strict preference, and the relationship [math]\displaystyle{ \succcurlyeq }[/math] as weak preference. Then

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

Such preferences might be captured by a utility function [math]\displaystyle{ u() }[/math], in which case

• the upper contour set of [math]\displaystyle{ x }[/math] would be the set of all [math]\displaystyle{ y }[/math] such that [math]\displaystyle{ u(y)\ge u(x) }[/math],
• the strict upper contour set of [math]\displaystyle{ x }[/math] would be the set of all [math]\displaystyle{ y }[/math] such that [math]\displaystyle{ u(y)\gt u(x) }[/math],
• the lower contour set of [math]\displaystyle{ x }[/math] would be the set of all [math]\displaystyle{ y }[/math] such that [math]\displaystyle{ u(x)\ge u(y) }[/math], and
• the strict lower contour set of [math]\displaystyle{ x }[/math] would be the set of all [math]\displaystyle{ y }[/math] such that [math]\displaystyle{ u(x)\gt u(y) }[/math].

Complementarity

On the assumption that [math]\displaystyle{ \succcurlyeq }[/math] is a total ordering of [math]\displaystyle{ X }[/math], the complement of the upper contour set is the strict lower contour set.

[math]\displaystyle{ X^2\backslash\left\{ y~\backepsilon~y\succcurlyeq x\right\}=\left\{ y~\backepsilon~x\succ y\right\} }[/math]

[math]\displaystyle{ X^2\backslash\left\{ y~\backepsilon~x\succ y\right\}=\left\{ y~\backepsilon~y\succcurlyeq x\right\} }[/math]

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

[math]\displaystyle{ X^2\backslash\left\{ y~\backepsilon~y\succ x\right\}=\left\{ y~\backepsilon~x\succcurlyeq y\right\} }[/math]

[math]\displaystyle{ X^2\backslash\left\{ y~\backepsilon~x\succcurlyeq y\right\}=\left\{ y~\backepsilon~y\succ x\right\} }[/math]

See also

• Epigraph
• Hypograph

Bibliography

• Andreu Mas-Colell, Michael D. Whinston, and Jerry R. Green, Microeconomic Theory (LCC HB172.M6247 1995), p43. ISBN 0-19-507340-1 (cloth) ISBN 0-19-510268-1 (paper)