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.

Given a relation on pairs of elements of set ${\displaystyle X}$

${\displaystyle \succcurlyeq ~\subseteq ~X^{2}}$

and an element ${\displaystyle x}$ of ${\displaystyle X}$

${\displaystyle x\in X}$

The upper contour set of ${\displaystyle x}$ is the set of all ${\displaystyle y}$ that are related to ${\displaystyle x}$:

${\displaystyle \left\{y~\backepsilon ~y\succcurlyeq x\right\}}$

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

${\displaystyle \left\{y~\backepsilon ~x\succcurlyeq y\right\}}$

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

${\displaystyle \left\{y~\backepsilon ~(y\succcurlyeq x)\land \lnot (x\succcurlyeq y)\right\}}$

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

${\displaystyle \left\{y~\backepsilon ~(x\succcurlyeq y)\land \lnot (y\succcurlyeq x)\right\}}$

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

${\displaystyle \succ ~=~\left\{\left(a,b\right)~\backepsilon ~\left(a\succcurlyeq b\right)\land \lnot (b\succcurlyeq a)\right\}}$

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

${\displaystyle \left\{y~\backepsilon ~y\succ x\right\}}$

and the strict lower contour set of ${\displaystyle x}$ is

${\displaystyle \left\{y~\backepsilon ~x\succ y\right\}}$

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

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

Consider a real number ${\displaystyle x}$, and the relation ${\displaystyle \geq }$. Then

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

Consider, more generally, the relation

${\displaystyle (a\succcurlyeq b)~\Leftarrow ~[f(a)\geq f(b)]}$

Then

• the upper contour set of ${\displaystyle x}$ would be the set of all ${\displaystyle y}$ such that ${\displaystyle f(y)\geq f(x)}$,
• the strict upper contour set of ${\displaystyle x}$ would be the set of all ${\displaystyle y}$ such that ${\displaystyle f(y)>f(x)}$,
• the lower contour set of ${\displaystyle x}$ would be the set of all ${\displaystyle y}$ such that ${\displaystyle f(x)\geq f(y)}$, and
• the strict lower contour set of ${\displaystyle x}$ would be the set of all ${\displaystyle y}$ such that ${\displaystyle f(x)>f(y)}$.

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

${\displaystyle (a\succcurlyeq b)~\Leftarrow ~[f(a)\leq f(b)]}$

though such definitions would tend to confound ready understanding.

In the case of a real-valued function ${\displaystyle f()}$ (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

${\displaystyle (a\succcurlyeq b)~\Leftarrow ~[f(a)\geq f(b)]}$

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

${\displaystyle [(a_{1},a_{2},\ldots )\succcurlyeq (b_{1},b_{2},\ldots )]~\Leftarrow ~[f(a_{1},a_{2},\ldots )\geq f(b_{1},b_{2},\ldots )]}$

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

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

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

• the upper contour set of ${\displaystyle x}$ would be the set of all ${\displaystyle y}$ such that ${\displaystyle u(y)\geq u(x)}$,
• the strict upper contour set of ${\displaystyle x}$ would be the set of all ${\displaystyle y}$ such that ${\displaystyle u(y)>u(x)}$,
• the lower contour set of ${\displaystyle x}$ would be the set of all ${\displaystyle y}$ such that ${\displaystyle u(x)\geq u(y)}$, and
• the strict lower contour set of ${\displaystyle x}$ would be the set of all ${\displaystyle y}$ such that ${\displaystyle u(x)>u(y)}$.

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

${\displaystyle X^{2}\backslash \left\{y~\backepsilon ~y\succcurlyeq x\right\}=\left\{y~\backepsilon ~x\succ y\right\}}$

${\displaystyle X^{2}\backslash \left\{y~\backepsilon ~x\succ y\right\}=\left\{y~\backepsilon ~y\succcurlyeq x\right\}}$

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

${\displaystyle X^{2}\backslash \left\{y~\backepsilon ~y\succ x\right\}=\left\{y~\backepsilon ~x\succcurlyeq y\right\}}$

${\displaystyle X^{2}\backslash \left\{y~\backepsilon ~x\succcurlyeq y\right\}=\left\{y~\backepsilon ~y\succ x\right\}}$

• Epigraph
• Hypograph

• 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)